As for why I support this, I think it makes sense diplomatically to have an apolitical head of state and I also believe the British monarch functions similarly to second amendment in the US. If the UK ever needed to revolt against the government the monarch serves as way in which the public (and armed forces) could side with the state over the government. I cannot express how much I like that our armed forces swear allegiance to a largely powerless Monarch that represents the state rather than Rishi Sunak.
However, getting back on topic here, the fact the UK has a monarch in my opinion has zero relevance to the fact the UK is becoming a place increasingly hostile to protest. Instead, it seems to me to be far more to do with the fact our two major political parties both have authoritarian tendencies, which I suspect stems from the UK having cultural values which lend themselves to certain kinds of authoritarian social policies.
Specifically, laws around limiting "improper" behaviour like rowdy protests, being rude online, and consuming porn that's a little too edgy seem to be ways in which our dislike of improper behaviour manifest politically. In most ways we're quite a liberal society, but there are certain ways that our conformist and overly polite nature seems to work its way into our politics via democracy – protest perhaps being the most clear example of something that people here seem to struggle with since freedom of expression requires an acceptance of impoliteness.