The thing is Republicanism won't actually solve the problem you mention, of the core group of powerful MPs being overwhelmingly representative of the elite. If anything it will swing things further under their control.

The monarchy aren't that rich or powerful at all by modern standards. The true wealth is in random parts of the aristocracy that owns massive parts of west London, but again merely turning into a republic is not going to result in that being redistributed any more palatably than it is today.

It is a dated institution, and it is a bit odd to have people born into such a thing, but it really isn't that actively harmful, and certainly less harmful than a headlong rush into a worse alternative when it appears to be slowly diminishing into irrelevance without the need of any external stimulus.