PDF reader that refreshes at updates and stays put
The answer was in a comment by Jonathan Schuster to the question Reopen PDF in Preview on the same page when modified externally that Graham Miln pointed at.

In his comment, Jonathan wrote


  For what it's worth, I've found Skim to be a good viewer that doesn't
  have this issue. Just make sure to check "Check for file changes" in
  the Sync section of Preferences.


which neatly answers my question.