The main issue with the quoted example is that GET is intended for non-destructive interaction with no side-effects. Whether you use PUT or POST is a much smaller argument than whether to use GET or something else.
To most people, honking the car's horn is an unwanted side effect.
As has already been pointed out, some browsers will fetch-ahead to ensure that pages appear to load instantly. If you open a page which is a bunch of buttons which do stuff like honk the horn, lock the door, raise the window or open the charging port door, there will be unwanted side effects.
So yes, we do care that much.