Why does it need to be large to function independently? There are a bunch of far smaller nations in the EU/Europe: Luxembourg, Andorra, San Marino, Liechtenstein, etc. If Northern Ireland became independent, it could just join the EU (regular Ireland is also an EU member), so they wouldn't have to maintain their own currency at least, and would benefit from the free trade. If Luxembourg can function as it is, I don't see why Northern Ireland can't do the same.

Which is ironic given how long foreign rule has had to sort the place out.

Not really. Northern Ireland is quite small.