Having seen how it's done via plenty of news reports, having a swap stuck up the back of my nose certainly looks plenty invasive, not to mention potentially painful.

I can certainly imagine someone who's asymptomatic and is more nervous about it being disinclined to voluntarily undergo testing.

If it were a finger-tip prick or cheek-swap it'd probably have a lot more acceptance.

This is true, very few would object to a finger-prick, and there was talk of a "spit test" a few weeks back; that would push testing rates to close to 100%