I think it has less to do with shyness and more the fact that almost nobody speaks to pollsters on either side of the isle. Most polling is done by phone call. When was the last time you answered a call from an unknown number?

I don't think Republican voters are shy.

It's just incredibly hard to build a representative sample of the population.

It absolutely does. You can generally count on anyone who describes themselves as "Centrist" or "Apolitical" (doubly so if you're on a dating site) to be more to the right.

It used to be "Libertarian" which for a subset was "I'm a Republican who likes to smoke weed".