So,it is second roman republic versus woud-be-third rome?
No, it's a fight over Russian expansionism. America *isn't* a "Second Roman Republic" and has zero links to it. The only person of note/fame who thinks otherwise is Ben Shapiro, and he's not only way outside mainstream American political thought but he's also outside mainstream American *conservative* political thought. Hell he's not even Christian, much less Orthodox Christian, so his obsession over who is Rome's "successor" is even weirder considering that's a thing only the Orthodox care about -most American Christians are either Catholic or Protestant (that is, offshoots of Catholicism) and so don't believe Rome *has* a successor.
Then the Venetians started fighting wars with the Byzantines
during the fourth crusade.
Afterwards, particularly during the Turkish invasion the Venetians poached lots of talent from Constantinople, and one theory is that the scholars, artisans and artists they got helped incubate the Renaissance.
So you can say that the Byzantines are indirectly responsible for both the 3rd and 4th Rome.
Also,
proto-Protestantism thanks to the Bogomils
It's more like the Byzantines got pulled into a slap fight between the Italian city states (specifically, Genoa, Lisa, and Venice) that was ultimately over money. The Genoans grabbed the Pisans and rioted through the Venetian quarter of Constantinople and killed/injured a bunch of people. They in turn (rightly) got booted from the city for being assholes, but it had the unintended effect of allowing the Venetians to entrench themselves in the city as the dominant merchant faction (partly due to the resources and links Venice could draw on but the Greeks couldn't). This, coupled with hatred for Catholics over the Great Schism, meant that when a palace coup was executed by the Greeks (something that was normal for them but notably *not* normal in the West) and targeting Catholics was used as a way to get domestic support by the new guy (Romanus IV, I believe) it blew up in spectacular fashion and pissed off a LOT of people (murdering old people and children while selling most everyone else into slavery will do that).
The funny thing is, Venice and the rest weren't actually in a position to do anything about it until Yet Another Claimant to the throne showed up asking for help while saying "Help me with this and I'll pay off the debts you guys worked up setting up the Fourth Crusade" -and Venice went for it because they had, in fact, gone deep into debt prepping for what they were told was going to be a massive venture. So off they all went, and did it...only to find out that the Byzantines had, in fact, been inadvertently stringing them along because the claimant in question (Alexios Angelos, later Alexios IV) turned out to have no idea what he was talking about. The Westerners, whose ideas of governance, inheritance, etc were quite different from the Greeks, and whose faith made them hated, kept trying to push for the bargain they'd made to be upheld, whereupon the Greeks reacted poorly...only to have their asses handed to them because commoners and peasants going up against trained soldiers generally goes very poorly for the former, as it did here.
The Crusaders, sick of this shit, overthrew the Empire and set up their own regime. That is another story in and of itself but it basically was the beginning of the end for the Empire.
So it's less the Venetians having multiple beefs with the Byzantines and more just a bunch of angry Italians who were pissed off at a perception of being cheated out of money they earned and having the muscle to retaliate.
As for "poaching" artisans...it's less that and more that Venice was one of the first major Western cities that wasn't a backwater of note refugees came to while fleeing, trade and travel routes being what they were at the time. The other contenders were generally Genoa (for the same reasons as Venice) and Florence (which had conquered the *third* major maritime player, Pisa, about 50 years before).
You are correct, however, in that this is generally considered a significant factor if not the biggest factor in kick-starting the Renaissance.
This all has about as much to do with the current geopolitical climate as Columbus's voyages do...that is, there is no straight line between the two. Thinking otherwise is a bunch of weird strings of thought, a *massive* oversimplification, and missing the forest for a particular and not particularly noteworthy tree.