The great schism between the orthodox and catholic churches isn't like the schism between the protestants and Catholics. The protestant and catholic split is rooted in a fundamental disagreement on what the religion is; historically there has been conflict intertwined with politics too, but (Ireland notwithstanding) that isn't relevant to the Catholic/Protestant divide.
It is true that certain protestant sects are effectively "the catholicism we have at home [in england]" and you are right that those probably can be convinced to rejoin the catholic church but the majority of protestant sects have a firmly-rooted belief that the church is an organization created by humans to worship god and there is nothing inherently sacred about it. They also tend to reject anything outside of the old and new testaments compiled at nicea as being canon.
There's a fascinating bit of cognitive dissonance wherein they believe that God is still actively involved in the world and has been for the past 2000 years yet they haven't made any attempts at recording them; I think the logic is that they'd need the church to have some sort of divine authority to add to the bible and they've already ruled out the church having that authority so the bible is effectively set in stone forever. But that's irrelevant, I'm getting off-topic here.
Anyways, as far as unification goes it doesn't really matter that nobody knows or cares about ancient wars between catholic and protestant kingdoms and it doesn't matter that they can all get along and be neighbors and even have their churches work together on charity projects because the schism between the catholics and protestants is rooted in ideology not animosity. There's no compromise between the pope being a direct line of succession from peter and the pope being "just a guy in rome who makes great sermons" and I can't imagine they're going to want to take 1700 years of catholic lore and add it into their canon like its no big deal either.
Another roadblock is that the protestants themselves are highly fractured, often due to minor disagreements over pedantic minutiae that at least 99% of their members don't care about (IIRC one of the disagreements was over whether Jesus meant it literally when he said the bread and wine are his flesh and blood or rather that was a figure of speech, i think the calvinists and the lutherans are on opposite sides of that disagreement) but they've all had a long history of peaceful cooperation and they've never let that turn into an actual conflict yet still they never even try to unite. They don't see any point as long as they can coexist peacefully as separate churches because the only thing that would grant them is consolidation of power, which they are largely disinterested in. So even putting the ideological debates and factionalism aside, they'd need to be convinced that there is even a point in unifying with the catholic church when they can continue to peacefully coexist as separate organizations.