My feeble attempt to fix cookie paths has apparently had no effect. The logon processes appear to work. However, I am apparently not recognized as logged-on at Spanner Wingnut and/or the comment form simply doesn't come up. The ceremony succeeds, but the result doesn't occur.
When I try this:

This appears to work:

And so does this:

I can see the cookies. If I delete the cookies they reappear. I tried republishing what I could but that makes no difference.
If I enter a bogus user name and password, I also receive the appropriate error message. On password recovery, I notice that there is only one e-mail address on file. It's not my usual one but it is the one that I have messages reporting Movable Type activity use. But the full ceremony, including notification that there has been a password recovery request, works fine. (I ignored the request, and everything works as expected.)
I can also register a new user, not with an user ID already used, and that ceremony, including e-mail confirmation, works just fine:
The common factor is that the comment form simply doesn't come up. I am fairly certain nfoCentrale Plan B is called for: creation of cgi-bin/mt/ in all anchored sites that have Movable Type blogs.
Before enjoying the Plan B pain, I will make one more change on the off-chance that the simplest thing, however inplausible, might work.
Here's a simple change to mt-config.cgi that I should probably make anyhow:
We'll just have to see what happens. Stay tuned.
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One problem, not discovered until working on this problem, is that I have been putting ## comments at the right of non-comment lines. That doesn't work. It is only the # in the first column that is recognized as commentary. In some cases, the comment will be taken as part of the parameter information, and different things can happen. There may be white-space issues too.
I now don't know what would have improved this situation, since I didn't figure out this particular problem until I had made some other significant changes.
I will stick with those changes because they are also recommended as part of reducing the threat surface of Movable Type installations. I avoided them at first because I didn't want to muck around in the web-hosting account using GNU/Linux command-line operations. It was easier than I feared, so I can do that little much for other sites now.