Prior to this commit, the `HttpEntityMethodProcessor` would avoid
writing ETag/Last-Modified response headers before calling
`ServletWebRequest` to process conditional requests. This was done to
avoid duplicate response header values due to headers being already
written to the underlying servlet response.
This is still necessary for GET/HEAD requests, since this is properly
handled by `ServletWebRequest` for those cases. But
`HttpEntityMethodProcessor` should not make that decision for
PUT/PATCH/POST responses since developers are adding response headers on
purpose and should be in control of the situation — whereas
`ServletWebRequest` does not write those headers in those cases.
Issue: SPR-14767
ee17f56626