Apologies for the amateur's question, but do coils need the wrapper for their functionality or are they just for looks? I noticed a couple of my TZ coils  are bare, ie have no paper around them and I'm wondering if it's a problem.

cannot see a problem apart from the paper could offer a little protection. The big issue is that you do not know what type of coil is installed unless you're great at visually assessing the wire gauge and the number of turns/windings. So at the moment you are reliant on the fact that he correct coil is mounted.

And yes I suppose a wrapper looks better and more original.

I hope my advise is pretty much correct, otherwise I am sure someone will fix that.