That's basically the Republican argument

(note that Republican in this context is very different from what Republican means in US politics)

Personally, I'm not too fussed about such things, and I would pay more attention to the republican cause if they moved from being negative about the institution to discussing what sort of constitution they would want in place of the Monarchy, because we do still need a lot more checks and balances in UK politics and to abolish the Monarchy without addressing this would be a colossal waste.