In that instance, the King refused to acknowledge the law as it did not apply to him, he was duly executed under hastily put together legislation and within 15 years everyone who manufactured that legislation was either dead or executed for treason. The Duke of Edinburgh was indeed breathalysed but he isn't the monarch - and we didn't get to see what would've happened if he was over the limit. I suspect we never would.
I've argued for equality under the law - and your rebuttal is that the Monarch doesn't get to be part of society in the same way as everyone else. They are essentially a gilded national pet, unable to express any opinions and you're right, fewer civil rights. I think that's a terrible thing to do to a person, regardless of the compensation you might give them. What you have suggested is correct, it's a form of indentured servitude in perpetuity based solely on tradition. That doesn't make it ethical and you've essentially bolstered my argument.
I heard a rumour, that the Queen had a little cottage somewhere where she enjoyed doing things like washing up and cooking - play acting at being a normal person. I found that profoundly sad. I thought it was terrible that two teenage boys were made to march behind the coffin of their mother in front of the world's press. I think the lack of equality I find abhorrent travels in both directions.
On a personal level, I liked Mrs Windsor, I wanted her to have some agency in her own life. :o)