Charles exerted lots of control behind the scenes while he was Prince of Wales, lobbying, receiving lobbyists, getting inside information, accepting huge donations, etc.
Influence is not the same as authority. I'm sure he has influence - but then so do lots of people. And at least as public figure he has a degree of visibility. Governments don't exist in a vacuum.
This is true, and yet there are plenty of other unelected people with far more influence than him. The US also has lots of rich, powerful and unelected people who exert vastly more political influence than the average citizen. That's not to say that I approve of Charles's lobbying activities, but he's hardly the worst example.
There's no 'intended message' to be found there, just some isolated incidents of dumb police officers arresting people for silly reasons (as acknowledged by their subsequent 'de-arrest').
Police arresting political dissidents in the name of an ostensibly powerless monarch is just one of those quirky things that happens in real democracies. Nothing to be concerned about.
Police can arrest people at their discretion. If these people were actually being prosecuted, I'd see your point. Otherwise, yes, it was wrong to arrest them, but you can find people being arrested for stupid reasons in any country. It doesn't require any kind of establishment conspiracy for that to happen. If you are raising a legitimate issue here, it's the broad powers of arrest that police have, not anything to do with Britain being a monarchy rather than a republic.