Loyalty oaths caused a lot of agitation from civil liberties groups but someone really devoted to overthrowing the established order at all costs might just take the oath to worm their way into a position where they could do some real damage.
But two qualifications: 1) if you were a dedicated Communist or fascist or anarchist or whatever you'd look like a hypocrite to your friends if you took the oath, so you would be much less likely to vote or serve in the government, and 2) some oaths required you not just to promise to honor the Constitution, but to promise that you'd never been a member of the Communist party or other subversive organization -- if you lied presumably you could be taken to court for it.