SOLVED
It took much more time and effort than expectable, but Shane wouldn't let it go till identified and solved. Many-many thanks and admiration again.
Re-formulation for clarity (November 3):
1. I type any program name, e.g. cmd, notepad, etc. in the search pane.
a. If I double left-click on it, it starts to run.
b. If I right click, then choose the menu item 'Run as Administrator' nothing happens.
2. I go to any program’s, e.g. cmd, notepad, etc., exe file in the respective program folder, and right click, then choose the menu item 'Run as Administrator' it starts to run as administrator.
That is, the issue is with merely the shortcut.
I'm not aware of anything else that can go wrong with a shortcut than its registration. Of course, a. anything may exist beyond my rather limited knowledge; b. even if I'm right in that, I have no idea which registration key to change and how.
I hope for one of these kinds of answers:
1. Oh, it's not registration, go to this place in control panel and change this preference; or: download and run this utility, etc., or
2. Yes, it's registry issue, change this registry key this way.
(end of revision)
If I right click a program (cmd, notepad, etc.) from start or start/search {corrected: and choose 'Run as Administrator'} nothing happens. I can run anything as admin directly from the exe files though.
Added later:
I clicked 'open file location', got to AppData/Roaming/Microsoft/Windows/Start Menu/Programs and experienced the same malfunction. When, however, in case of cmd, clicked 'open file location' again, I got to the System32 folder, and here I found OK;
i.e., the shortcuts to AppData/Roaming/Microsoft/Windows/Start Menu/Programs seem to be the culprit. Possibly their registration? (end of addition)
sfc/scannow or Dism.exe /Online /Cleanup-Image /Restorehealth don't help and I can't find which option to choose from Tweaking's repair.
Thanks a lot.