A Republic would not require an active Head of State such as in France or the USA, it could be purely ceremonial, or even got by without I suspect. Although the benefits of actually doing seem would seem to be more symbolic than actual.

If the Americans are so keen on a monarch they could seek to rejoin the Commonwealth and share ours, since the end of the Stuart line I don't think the title is contested anymore. Their constituntional separation of religion and state may present problems as the Queen is also Head of the Church though, as might the bar on Catholics. To be honest I haven't heard much clamour from my American collegues though, being a Republic seems to suit them just fine.

I am not sure what the connection you seem to see with the EU is, several other EU states are monarchies, and they seem to manage ok. Given that the DoE also held titles in Denmark and Greece and the Queen herself is from the line of Hanover I can't think of a better advertisement for European integaration than the various Royal Families that used to span the continent.