I’m marking this as ‘solved’ with some caveats…
This very strange behaviour was actually creating conflicts all over the place, namely, for some very weird reason, it managed to exhaust all system handles (file handles, socket handles, Internet socket handles, everything). Granted, the system didn’t stop working: the main daemons had already grabbed their fill of such handles and could happily work with the amount they got. Things would just start to misbehave when something new required a handle for some purpose — a typical example: making a DNS request, for instance. It would fail mysteriously. At the application level it wouldn’t be clear what had failed; the application would just be told that the operating system couldn’t create that particular socket (or file, or whatever).
Retrying a bit later would mysteriously succeed again… for a while… then stop working. This was a new situation, which I had never experienced before. Running out of file handles, well, yes, I had seen that once in a while — the errors are not so mysterious and clearer to read. Running out of sockets… well… I certainly expected a clearer error, being logged somewhere — a bit more than ‘authentication failed’ or ‘could not resolve IP address’ or ‘could not find file’.
Note that this is an oversimplification — there is actually something much deeper going on, namely, why some applications didn’t seem to be affected, while others (such as Cells or Postfix) do, and others still get partially affected, in the sense that they get an unknown error, wait a bit, a handle becomes available, retry, get a positive answer, and releases the handle for other processes to use. Things like, say, PHP-FPM, for instance, would gracefully deal with the situation — they might fail to respawn an instance to handle a request, but that would be ok, the monitor would just kill that instance (thus releasing a few handles back) and try again (with luck, this time it would work).
Miraculously it could be traced to the issue on this thread. It doesn’t seem to bear any relation whatsoever with it, but… once I removed the storage and created it from scratch without the former-directory-now-files entries, everything went back to normal. Everything.
Why? Well, I seriously suspect a relationship with the ‘other’ issue I reported: Cells 4.0.0 RC is constantly writing memory-mapped files to disk, even when nothing is happening - #2 by GwynethLlewelyn
It still baffles me, though 
But I’m glad that it’s ‘fixed’ in my case now (even though I’m not sure why), and maybe this thread might be useful to someone else in the future, who knows…