Nice explanation. I think the timeline laid out in Matthew 24 makes it pretty clear that "the tribulation of those days" comes before "our gathering to Him.":

21 For then there will be great tribulation, unequaled from the beginning of the world until now—and never to be equaled again.

22 "If those days had not been cut short, no one would survive, but for the sake of the elect those days will be shortened. 23 At that time if anyone says to you, 'Look, here is the Messiah!' or, 'There he is!' do not believe it. 24 For false messiahs and false prophets will appear and perform great signs and wonders to deceive, if possible, even the elect. 25 See, I have told you ahead of time.

26 "So if anyone tells you, 'There he is, out in the wilderness,' do not go out; or, 'Here he is, in the inner rooms,' do not believe it. 27 For as lightning that comes from the east is visible even in the west, so will be the coming of the Son of Man. 28 Wherever there is a carcass, there the vultures will gather.

29 "Immediately after the tribulation of those days" 'the sun will be darkened, and the moon will not give its light; the stars will fall from the sky, and the heavenly bodies will be shaken.'

30 "Then will appear the sign of the Son of Man in heaven. And then all the peoples of the earth will mourn when they see the Son of Man coming on the clouds of heaven, with power and great glory. 31 And he will send his angels with a loud trumpet call, and they will gather his elect from the four winds, from one end of the heavens to the other.

There is no doubt in my mind that that last sentence is the rapture (gathering) that Paul was referring to in 2 Thessalonians 2:1. It couldn't be more clear that the elect must go through the Greatest Tribulation before they are raptured.