Alas, the celebration was premature. Shortly after posting msg405488 above, I installed the updated 3.10.0 on yet another macOS system and, on this one, I observed a problem: under some circumstances at least, the Save dialog window becomes orphaned and does not disappear until IDLE terminates. Here is one sequence of commands I've used:
- launch IDLE -> IDLE opens an IDLE shell window
- press CMD-N to open a new IDLE edit window
- type Python text into the window with a syntax error:
   x = $1
   print(x)
- press CMD-S to save the text into a file -> file Save window opens
- enter a file name and click on Save -> file is saved and IDLE edit window regains focus
- press F5 to run the script -> "Invalid syntax" popup appears
- click on OK to dismiss popup -> the popup disappears and the edit window regains focus. BUT sometimes (and not always) the Save window reappears behind the edit window and does not respond to Cancel or Save buttons, remaining orphaned until IDLE is exited.

This happens on both Monterey and Big Sur, I haven't tried earlier systems. So, while the current fix does make IDLE file dialogs usable on Monterey, it introduces an orphaned (though seemingly harmless) window on presumably all systems.

The big question is why didn't we see this during earlier testing? In my case, the answer is that on all the systems I tested with, the Save dialog window was in its default "compact" form and thus was completely hidden by the IDLE windows. Plus it doesn't seem to always happen if you vary the sequence of events slightly, not surprisingly. I only noticed the orphaned Save window now because on this latest system the Save window defaulted to the fully expanded version (with the side bar etc) and that Save window was wide enough to not be fully hidden behind the other IDLE windows. Sigh!

Until we can resolve this, I have reverted the changes to the website so that the default macOS download is again the original 3.10.0 installer.  The patched installer is still downloadable for testing at https://www.python.org/ftp/python/3.10.0/python-3.10.0post1-macos11.pkg.