The text goes much further than "praying for all men to be saved" - it explicitly states that "God desires all men to be saved".
1 Timothy 2:1 Therefore I exhort first of all that supplications, prayers, intercessions, and giving of thanks be made for all men, 2 for kings and all who are in authority, that we may lead a quiet and peaceable life in all godliness and reverence. 3 For this is good and acceptable in the sight of God our Savior, 4 who desires all men to be saved and to come to the knowledge of the truth. 5 For there is one God and one Mediator between God and men, the Man Christ Jesus, 6 who gave Himself a ransom for all, to be testified in due time,
What did Jesus say?
Matthew 7:21-23
(21) Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.
(22) Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works?
(23) And then will I profess unto them, I never knew you: depart from me, ye that work iniquity.
What did Paul say about "all" being saved?
1 Corinthians 6:9-10
9 Know ye not that the unrighteous shall not inherit the kingdom of God? Be not deceived: [no wrongdoer] neither fornicators, nor idolaters, nor adulterers, nor effeminate, nor abusers of themselves with mankind,
10 Nor thieves, nor covetous, nor drunkards, nor revilers, nor extortioners, shall inherit the kingdom of God.
Galatians 5:19-21
19 Now the works of the flesh are manifest, which are these; Adultery, fornication, uncleanness, lasciviousness,
20 Idolatry, witchcraft, hatred, variance, emulations, wrath, strife, seditions, heresies,
21 Envyings, murders, drunkenness, revellings, and such like: of the which I tell you before, as I have also told you in time past, that they which do such things shall not inherit the kingdom of God.
Ephesians 5:5 For this ye know, that [no wrongdoer] no whoremonger, nor unclean person, nor covetous man, who is an idolater, hath any inheritance in the kingdom of Christ and of God.
1 Corinthians 3:17 If any man defile the temple of God, him shall God destroy; for the temple of God is holy, which temple ye are.
Twenty two categories of unrighteous people who have no inheritance in the kingdom of God.
1. adulterers, 2. covetous, 3. drunkenness, 4. effeminate, 5. emulations, 6. envious, 7. extortion 8. fornication, 9. hatred, 10. heresies, 11. homosexuals, 12. idolators, 13. lasciviousness, 14. murder, 15. reveling, 16. revilers 17. sedition, 18. strife, 19. thieves, 20. uncleanness 21. witchcraft. 22. Wrath
1 Corinthians 6:9-10, Galatians 5:19-21, Ephesians 5:5, 1 Corinthians 3:17