You are right, the only direct reference JWs literature makes concerning governments finally turning on religion immediately prior to Armageddon is found in Revelation where the wild beast turns on the harlot that rides on its back.
However, as you also know, they link the Harlot as representing religion by quoting scriptures in the OT. First they take you to Genesis where the tower of Babel caused false religion to spread worldwide. Then they make reference to ancient Israel's spiritual infidelity to God where such infidelity is described as harlotry.
They also quote the book of Daniel where beasts appear in a vision and are identified with a sequence of world powers. Then these are linked to the composite seven-headed beast in Revelation which the Harlot Babylon the Great, which they tag as the world empire of false religion whose riding the beast is described as its cooperation with world governments and its spiritual harlotry and voila! There you have it. When the so-called composite beast is attacks the harlot it therefore is really attacking the world empire of false religion.
Those with the mark of the beast are slaves of the beast and are those who support the beast.
Here is where the voting part emerges.
So to Bible students who have no background in the Bible it all seems 100% true and it becomes extremely difficult to give these symbols any other meaning except the one which seems clear from the scriptures themselves as explained by the WTBTS.
Actually, I am still reluctant to participate in voting because of this teaching.
What gives me pause is that Daniel Shadrac, Meshach, and Abednego had no qualms whatsoever in working for the government back then and if they indeed had thought it as being marked by the wild beast then they would not have done so.
It also seems irresponsible to remain neutral when a government is abusing the citizens and we might vote that government out of office and in that way alleviate suffering. I really don't see how that constitutes a rejection of God's Kingdom in favor of a solution by mankind. In fact, it is an act of compassion to the victims and compassion or empathy is a quality that the Bible repeatedly tells its readers is very important to express.
I don't see how standing by with arms crossed while millions are being abused and exterminated in concentration camps and simply praying for a day when all will be under God's rule is a virtue. It could very well be viewed as callousness toward suffering of others or tacit cooperation with the wicked.
So despite all the biblical references there are ethical problems that emerge and which bring to mind what Jesus told the Pharisees "I want mercy and not sacrifice!"