This is weird. As I went through the update process to EE 2.5.3, I noticed at some point (before finishing) that the problem had taken care of itself. I then happily completed the update process, including updating to MSM 2.1.4. But when I checked again the problem had reemerged.
I have no idea why this could have happened, because all I did to the site was to follow the update instructions exactly as described in the EE docs. In an attempt to backtrack to a state where things had mysteriously just started working again, I restored the old index.php and admin.php files on the MSM sites in question, and reinstalled MSM 2.1.3. This should have taken me to the point where the sites had begun working normally, but it didn’t help. I’ve now reinstalled 2.1.4 and re-updated the index.php and admin.php files, so I’m all updated.
Any idea why for a brief period there the problem disappeared, and then suddenly came back again? Hopefully that offers some kind of clue as to what’s going on, though it’s beyond me to guess what that may be…