Internal: refactor copy headers mechanism to not require a client factory

With #7594 we replaced the static `BaseRestHandler#addUsefulHeaders` by introducing the `RestClientFactory` that can be injected and used to register the relevant headers. To simplify things, we can now register relevant headers through the `RestController` and remove the `RestClientFactory` that was just introduced.
Not sure I get this...this set if mutable, cause you can potentiallly register relevant headers all the time through `registerRelevantHeaders` method. Unless we find a way to "seal" the headers after having set them, I think copy on write was better? What am I missing?