The monarchy is an old institution, a bit like an old historic city, cathedral or museum. The British are very distrustful of knocking down old things to replace them with new because once they're gone they cannot be brought back instead we just generally tweak/evolve them to suit the modern age without completely replacing them.

Would getting rid of the monarchy actually really change anything other than being change for the sake of change? Not really, so what would be the point?