An election can mean that half the country hates you. Sometimes more than half, since there are more than two parties in Britain, and it happens that someone whose party gets fewer popular votes can win more seats in Parliament and make you PM. And, of course, Britons don't vote for the PM directly, though even if they did, he'd still have his opponents.
I suppose the British think that getting rid of the monarchy would leave a large vacuum in their culture and take away a stabilizing influence in their politics. Also, it's good for tourism.
Prime Ministers are not voted on directly. If that were the case, Churchill would have wiped the floor with Atlee in 1945, but it’s the majority Party in Parliament that chooses the PM.