For me it's more the out dated ceremonies and the displays of wealth that I don't like. I think if the monarch just lived in a nice country house and deprecated silly things like the Crown Jewels and the Royal Carriage then I'd be completely fine with them.
I think it's the "royal" part of the monarchy that people oppose rather than the system itself.
Yes they’re wealthy and have exercised power that wealth gives you but they’d still be wealthy if they were not a monarch.
If we want to talk about the power that money brings you in politics then we’re having a different discussion, a discussion that is not UK specific.
But yes the monarchy as an institution has not involved itself in political matters in living memory, and has not exercised political power (executive orders, vetoing bills) in centuries.
Parliament is the sovereign political power in the UK. If they wanted to they could pass a bill today that dismantled the crown and the monarchy.
[1] https://www.theguardian.com/uk-news/ng-interactive/2015/may/...
[2] https://www.theguardian.com/uk-news/2022/jul/14/queen-immuni...
I think if you made a list of how many laws and bills have been influenced by private lobbying and donations you’d have an inconceivably large list.
My point earlier is that the wealthy get their say, the monarchy as an institution hold the same amount of political power as the wealthy in the UK, but they do not hold direct political power.
An elected head of state will be impossible to keep politically neutral and inevitably will polarise things.
Maybe a system with an appointed head of state instead of elected, similar to peerages in the House of Lords. Someone vetted and trusted, approved by an elected parliament.
Although the Monarchy is already completely controlled by parliament which is democratically elected so I’m not sure what the difference would be.