It is Calvinism - the L of TULIP is for limited atonement - that the cross did not work for all, and that those it did work for cannot avoid it.
I find the notion problematic for a number reasons:
1) As you have said, what is the point of our evangelism, taking the Gospel to all nations, the need to hear to believe in Romans 10 etc.
This is not actually a problem historically. Calvin, by the way, did not invent these doctrines. These were the Reformation doctrines as expressed by Luther himself, which Calvin only gets his name attached since he was so influential later. And before Luther was Augustine (not surprising, since Luther was himself an Augustinian monk), and before Augustine, of course, was the scripture, and the Apostles.
We who are Christians desire to serve God, and do so because He commands and pulls us to it. John Knox, that student of Calvin and of Geneva, was not a toothless evangelizer, nor were any of the other great reformers who went about the world with their evangelistic zeal.
2) There are other verses which seem to contradict it:
There are no verses which contradict it, only verses which we, English speakers, do not understand. One must be aware of the unique peculiarities Jews make use of in their language. For example, from Gill’s commentary, commenting on the Jewish use of the words “all the world, the world, the whole world, all the men of the world,” etc, Gill begins by quoting Jewish examples where the word is used in radically different ways from how we would use it today:
“Nothing is more common in Jewish writings than to call the Gentiles, “the world”; and “the whole world”; and “the nations of the world” (l); See Gill on John 12:19; and the word “world” is so used in Scripture; see Joh_3:16; and stands opposed to a notion the Jews have of the Gentiles, that “there is no propitiation for them” (m): and it is easy to observe, that when this phrase is not used of the Gentiles, it is to be understood in a limited and restrained sense; as when they say (n),
‘it happened to a certain high priest, that when he went out of the sanctuary, “the whole world” went after him;’’(n)
which could only design the people in the temple. And elsewhere (o) it is said,
“amle ylwk, “the “whole world” has left the Misna, and gone after the “Gemara”;’’
which at most can only intend the Jews; and indeed only a majority of their doctors, who were conversant with these writings: and in another place (p),
“amle ylwk, “the whole world” fell on their faces, but Raf did not fall on his face;’’
where it means no more than the congregation. Once more, it is said (q), when
“R. Simeon ben Gamaliel entered (the synagogue), “the whole world” stood up before him;’’
that is, the people in the synagogue: to which may be added (r),
“when a great man makes a mourning, “the whole world” come to honour him;’’
i.e. a great number of persons attend the funeral pomp: and so these phrases, “the whole world” is not divided, or does not dissent (s); “the whole world” are of opinion (t), are frequently met with in the Talmud, by which, an agreement among the Rabbins, in certain points, is designed; yea, sometimes the phrase, “all the men of the world” (u), only intend the inhabitants of a city where a synagogue was, and, at most, only the Jews: and so this phrase, “all the world”, or “the whole world”, in Scripture, unless when it signifies the whole universe, or the habitable earth, is always used in a limited sense, either for the Roman empire, or the churches of Christ in the world, or believers, or the present inhabitants of the world, or a part of them only, Luk_2:1; and so it is in this epistle, 1Jo_5:19; where the whole world lying in wickedness is manifestly distinguished from the saints, who are of God, and belong not to the world; and therefore cannot be understood of all the individuals in the world”
(From Gill’s Commentary on 1 John 2:2, quoting from (l) Jarchi in Isa. liii. 5. (m) T. Hieros. Nazir, fol. 57. 3. Vid. T. Bab. Succa, fol. 55. 2. (n) T. Bab. Yoma, fol. 71. 2. (o) T. Bab. Bava Metzia, fol. 33. 2. (p) T. Bab. Megilla, fol. 22. 2. (q) T. Bab. Horayot, fol. 13. 2. (r) Piske Toseph. Megilla, art. 104. (s) T. Bab. Cetubot, fol. 90. 2. & Kiddushin, fol. 47. 2. & 49. 1. & 65. 2. & Gittin, fol. 8. 1. & 60. 2. (t) T. Bab. Kiddushin, fol. 48. 1. (u) Maimon. Hilch. Tephilla, c. 11. sect. 16.)
Scriptural examples of this include:
Luk 2:1 In those days a decree went out from Caesar Augustus that all the world should be registered.
By this Luke means, not the whole habitable world, as much of it was still undiscovered, and not even the whole known world, which even in those days was not all under the power of the Roman empire, but just the Roman empire itself, or perhaps Judae.
Another one, which Gill mentioned:
1Jn 5:19 We know that we are from God, and the whole world lies in the power of the evil one.
Obviously we are no longer under the power of Satan, but have been released from his clutches by the power of Jesus Christ.
Another:
Luk 11:42 “But woe to you Pharisees! For you tithe mint and rue and every herb, and neglect justice and the love of God. These you ought to have done, without neglecting the others.”
Some translations render this “all manner of herb,” but, literally, it is “every herb,” which, obviously, the meaning is only every kind of herb.
St. Augustine uses this example in his explanation of the verse 1 Ti 2:4 “Who will have all men to be saved”:
“Or, it is said, “Who will have all men to be saved;” not that there is no man whose salvation He does not will (for how, then, explain the fact that He was unwilling to work miracles in the presence of some who, He said, would have repented if He had worked them?), but that we are to understand by “all men,” the human race in all its varieties of rank and circumstances,—kings, subjects; noble, plebeian, high, low, learned, and unlearned; the sound in body, the feeble, the clever, the dull, the foolish, the rich, the poor, and those of middling circumstances; males, females, infants, boys, youths; young, middle-aged, and old men; of every tongue, of every fashion, of all arts, of all professions, with all the innumerable differences of will and conscience, and whatever else there is that makes a distinction among men. For which of all these classes is there out of which God does not will that men should be saved in all nations through His only-begotten Son, our Lord, and therefore does save them; for the Omnipotent cannot will in vain, whatsoever He may will? Now the apostle had enjoined that prayers should be made for all men, and had especially added, “For kings, and for all that are in authority,” who might be supposed, in the pride and pomp of worldly station, to shrink from the humility of the Christian faith. Then saying, “For this is good and acceptable in the sight of God our Saviour,” that is, that prayers should be made for such as these, he immediately adds, as if to remove any ground of despair, “Who will have all men to be saved, and to come unto the knowledge of the truth” [I Tim. 2:1-4]. God, then, in His great condescension has judged it good to grant to the prayers of the humble the salvation of the exalted; and assuredly we have many examples of this. Our Lord, too, makes use of the same mode of speech in the Gospel, when He says to the Pharisees: “Ye tithe mint, and rue, and every herb” [Luke 11:42]. For the Pharisees did not tithe what belonged to others, nor all the herbs of all the inhabitants of other lands. As, then, in this place we must understand by “every herb,” every kind of herbs, so in the former passage we may understand by “all men,” every sort of men. And we may interpret it in any other way we please, so long as we are not compelled to believe that the omnipotent God has willed anything to be done which was not done: for setting aside all ambiguities, if “He hath done all that He pleased in heaven and in earth” [Ps. 115:3]. as the psalmist sings of Him, He certainly did not will to do anything that He hath not done.” (Augustine, Enchiridion on Faith, Hope and Love, Ch. 103. Interpretation of the Expression in I Tim. 2:4: “Who Will Have All Men to Be Saved”.)
Another example:
“As it is written, There is none righteous, no, not one: There is none that understandeth, there is none that seeketh after God.”
(Rom 3:10-11)
Obviously this cannot mean that no one seeks or understands, as, obviously, all Christians seek and understand (though maybe not perfectly). But, Paul’s meaning is more general,r referring to the depraved world, or of all Christians prior to Christ saving them, who, before that time, are incapable of seeing, believing or understanding until the Holy Spirit moves on them.
Another example, Christ distinguishing between the world and those given to Him out of the world:
Joh 17:9 I pray for them: I pray not for the world, but for them which thou hast given me; for they are thine.
Thus the world does not mean everyone in the world, but all those not given to Him.
Finally, more direct examples, by comparing seemingly contradictory verses with their parallels:
1Jn_2:2 And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world.
Joh_11:52 And not for that nation only, but that also he should gather together in one the children of God that were scattered abroad.
Thus the meaning is, Christ is the atonement for all the sins of the children of God scattered abroad, of every nation and tribe.
3) What does limited atonement say about God? The idea that God would allow people to be born knowing that most, (actually the vast majority if church attendance numbers are anything to go by) will end up in eternal conscious torment does not sound like the actions of a loving God to me.
You do not escape this problem, actually, as you would imagine that God allows millions of people to be born, in every century, into the world who are absolutely ignorant of the Gospel, and, likely, will always remain so, and therefore can never be saved. There is no salvation outside of Jesus Christ, nor is anyone saved by their works, as the scripture plainly teaches. They are called "without hope," and "without God" in the world, and all inexcusable and condemned by the law that has been implanted in their hearts. This is a righteous judgment, yet, in your system, God was powerless to give them the opportunity to be saved?
We do not claim that God "allowed" anyone to be born who would never be saved. We claim that He absolutely ordained it, as the all knowing and all powerful God, who foresees all things, does not see them contingently, but establishes the events of the world according to His own good purpose, which, though mysterious to us, ultimately will bring out good out of the evil that exists in the world all for our benefit.
"Thou wilt say then unto me, Why doth he yet find fault? For who hath resisted his will? Nay but, O man, who art thou that repliest against God? Shall the thing formed say to him that formed it, Why hast thou made me thus? Hath not the potter power over the clay, of the same lump to make one vessel unto honour, and another unto dishonour? What if God, willing to shew his wrath, and to make his power known, endured with much longsuffering the vessels of wrath fitted to destruction: And that he might make known the riches of his glory on the vessels of mercy, which he had afore prepared unto glory, Even us, whom he hath called, not of the Jews only, but also of the Gentiles?"
(Rom 9:19-24)
Tis a hard doctrine, but God is righteous, and is not obligated to give His mercy to all. But when He saves anyone, it is in mercy that He does so, as none of us are deserving of it; and if He judges us, it is in justice that He does so, as judgment is the only thing any of us actually deserve.
Rom 9:15 For he saith to Moses, I will have mercy on whom I will have mercy, and I will have compassion on whom I will have compassion.