Media Player stuck on 00:00 (randomly) when opened
I had to rebuild my phone after a firmware update; in doing so, I skipped over a lot of the Junk that I'd previously installed. The upshot is that I no longer suffer this problem (nor several others that had crept in). While it could be the software update that was the solution, I suspect that it was more likely that the clean install that did the trick.

On the plus side, the problem has disappeared; on the negative side, we don't know what the solution is.