...How do you account for Christ's bride being absent...
Note that the bride isn't absent from the tribulation, for there are no Christians outside of the bride; all believers form a single body (Ephesians 4:4-6), which is the bride of Christ (Ephesians 5:30-32). We Christians who will be in the tribulation are Christians after the cross and after Pentecost, saved in no different way than Christians who have died or will die before the tribulation. Many of us will be slain for the word of God and cry out to the Lord (Revelation 6:9-10); we will have washed our robes in the blood of the Lamb (Revelation 7:14); we will have the testimony of Jesus Christ (Revelation 12:17) and the faith of Jesus (Revelation 14:12) and will be in the Lord (Revelation 14:13); if we die, we will enter into heaven itself (Revelation 15:2); some of us will be beheaded for the witness of Jesus and for the word of God, and we will all live and reign with Christ (Revelation 20:4).
...during the time of His wrath?
How do you interpret 1 Thessalonians 1:10, 5:4-9...
Note that during the tribulation nobody in heaven says God's wrath "is come" until after the 7th trumpet (Revelation 11:15, 18), in the 7 vials of God's wrath (Revelation 15:1; Revelation 16), not one of which is poured out on those of us who have obtained salvation.
...one is taken while the other is left...
Note that Matthew 24:37-44 refers to the same "coming of the Son of man" as Matthew 24:29-31. There's no 3rd coming of Christ. Jesus is speaking to the same people in Matthew 24:15 that he is speaking to in Matthew 24:42.
I believe the ones "taken" at the 2nd coming are unbelievers who are killed, just as Noahs flood "took" evil men away (Matthew 24:39-40). The carcasses of the ones "taken" are eaten by eagles and other birds (Luke 17:36-37, Job 39:30, Matthew 24:28, Revelation 19:21).
I believe the ones "left" at the 2nd coming (Matthew 24:40, Zechariah 14:16) will be survivors of the heathen nations that came against Jerusalem, who will be forced to worship Jesus in the millennium (Zechariah 14:16-18), and whom we will rule "with a rod of iron" during that time (Revelation 2:26-29, 5:9-10, 20:4). They will be the ones who will populate the millennium.
The ones "gathered together" at Christ's 2nd coming are his elect (2 Thessalonians 2:1, Matthew 24:31). So I believe Christians won't be the ones "taken" or the ones "left," but the ones "gathered together."
...let us focus on evangelism rather than proper eschatology...
Note that these aren't opposed to each other. I believe preaching the rapture's timing is important because Jesus said it's possible for those who get saved to subsequently "be offended" by tribulation and "fall away":
"The same is he that heareth the word, and anon with joy receiveth it; Yet hath he not root in himself, but dureth for a while: for when tribulation or persecution ariseth because of the word, by and by he is offended" (Matthew 13:20-21).
"Then shall they deliver you up to be afflicted, and shall kill you: and ye shall be hated of all nations for my name's sake. And then shall many be offended, and shall betray one another, and shall hate one another. And many false prophets shall rise, and shall deceive many. And because iniquity shall abound, the love of many shall wax cold" (Matthew 24:9-12).
"They, which, when they hear, receive the word with joy; and these have no root, which for a while believe, and in time of temptation [peirasmos] fall away" (Luke 8:13). "Beloved, think it not strange concerning the fiery trial which is to try [peirasmos] you, as though some strange thing happened unto you: But rejoice, inasmuch as ye are partakers of Christ's sufferings; that, when his glory shall be revealed, ye may be glad also with exceeding joy" (1 Peter 4:12-13).