If you follow the order of events presented in Revelation chapter 16 (verses 16-21), the nations are gathered to "Armageddon" prior to the "great earthquake."
Thus, all the events listed from verses 17 through 21 (which includes 'Babylon's being handed the cup of God's wrath - v.19c) happen at "the place called in Hebrew, Armageddon." Saying that "Armageddon" occurs after Babylon's demise is technically incorrect. "Armageddon" doesn't "occur." "Armageddon" is a "place," a location (Greek topos - from which we get 'topography'). Events, such as Babylon's destruction and the "War" mentioned in 16:14 all occur 'at the place called Armageddon.'
Just as another point of note: Your argument depends on the 1995 WT understanding that Jesus ordering of the "goats" into the "everlasting fire" occurs midway through the great tribulation, that is, after "false religion" is destroyed. Note that in Matthew 24:14, after the preaching, "then the end will come." Verses 15-20 describe immediately prior to "the end." Then verses 21 and 22 describe great tribulation with 'no flesh being saved' except for God's intervening for his "chosen ones."
Having said all that, I can see where you are coming from in your reasoning. And if you were to accept the Society's speculative explanations, then I could see your conclusion also. What you are concluding would be a logical conclusion if you accept the Society's explanations as a premise. Oddly, though, I don't think the Society would agree to your conclusions. I believe the Society maintains that 'the door is shut' when the great tribulation begins, even though their explanation of the 'sheep and goats' parable would argue otherwise.
Take Care - And I did enjoy following your explanation. It has some 'outside-of-the-box-thinking' qualities to it.