I don't usually bump 'old' threads, however, this was the first one that came up when Googling for the issue I was having (which is, unsurprisingly, the same as the OP's..).
I've sorted the problem on mine and thought I'd share the solution here for others who find the same thread;
When I had set up 'Open With..' I had manually selected 'Media Centre 15' via the browse menu (and latterly v16). However, as the OP noted, there then two entries in the 'Open With..' panel - one that says 'Media Centre' and one that now says 'Media Centre 15/16'. Selecting the latter causes the issue the OP experienced. Selecting the former resolves the problem.
A quick check in the Registry shows that the former actually opens with 'MC16.exe' rather than 'Media Centre 16.exe' (I presume the same was true of v15).
So why did I manually choose the .exe file in the JRiver folder? Well when I upgraded from v15 to v16, I had only 'Media Centre 15' in the 'Open With..' menu, so I manually navigated to the .exe for v16. It was from there I started experiencing the problems with a double-click on a file not starting playback.
I've done a fresh installation and the problem is resolved. Hope this helps others who've experienced the same.