In history classes in Irish schools we were always taught that Wolfe Tone and the Ulstermen were inspired by the French and American revolutions to rebel against the British crown, but obviously there are substantial differences between these revolutions and between the thinkers who inspired them. So would I be correct in thinking these Protestant landowners would be more like the American founding fathers than the French revolutionaries or is there evidence to counter this?
The evidence to counter this is:
Their writings. United Irishmen like Wolfe Tone, Archibald Hamilton Rowan, Napper Tandy, Samuel Nelson, etc. were far more radical than Americans. Mostly because they did not have the same kind of material interests as American slaveowners did [e.g. they did not own slaves]
The United Irishmen knew they had to ally themselves with Catholic Ireland to win. Be it out of radicalism, empathy, or necessity, most United Irishmen were strong advocates and fighters for the dispossessed mass of Irish Catholics. Indeed, it is even misleading to think of the United Irishmen as a Protestant movement; there were hundreds of Catholics involved and it was mostly a Catholic rebellion.
As such, in my opinion, I think it is most likely that they would have established a secular republic. It's impossible to say how long it might have lasted, what it would have been like, etc., etc.
See: the scholarship of Kevin Whelan, Jim Smyth, and Breandán Ó Buachalla.