You're absolutely not wrong that this detracts from it. However, the courts in the US won't side with the policemen on this. They seem to be rather protective of the first amendment.

As an aside, police violence happens in the countries I mentioned. Not as much, but it does happen. Eg the yellow vest protests in France had a problem with this.

I really don't follow. The police doing something that the population doesn't support, doesn't mean the population supports the police actions. Not to mention "use of free speech" in this context is a big disingenuous. Are the police attacking people who are speaking their mind? Protesting and being asked to clear an area? Rioting? Peacefully protesting? Etc.

Free speech is extremely important in the United States, especially amongst sensible, regular everyday people. There are some elements on the far left who have been turning against free speech, but thankfully they are fringe groups that are largely ignored (Antifa/BLM Marxists, etc.)

Well, let's take an example : "It also avoids situations where you can be fined for insulting the president by holding up a sign that says "get lost, you prat" (France)."

The police fined the guy, which doesn't mean that the french people supported the police on this (they didn't), and that the law supported the police on this (it didn't).

So either police actions are a good scope to evaluate the importance of free speech, or either they aren't, but it shouldn't be only when it's convenient.