I guess I asked for it, because much older "PGP Desktop" had it already (via menu entry):

Maybe it's due to the fact that I used a non-admin installation? Actually I'm also surprised that it worked that way. What kind of debug logs could I supply?

In T5903#176176, @ikloecker wrote:

Please submit a separate feature request for this.

I'm late on the train, but (talking about Kleopatra) there is a menu entry to refresh all keys, but I miss a RMB (right mouse button) context menu for the list of certificates to refresh just the current one (much similar like the feature requested, but "one UI level higher").

OK, I'm sorry, please accept my apologies for not having read the dialog message carefully enough. It seems I ignore the second sentence, replacing it with the omnipresent "Do you want to continue?" in my mind. Maybe the excuse is that it was terribly hot and damp in my office, preventing me from thinking.

Strong objection to close: When answering the question with "no", installation aborts!

I checked it: There was an empty bin/gpgconf.ctl, and there still is.
Trying it again today, I still get error messages most notably about failed self-tests, but surprisingly the window is no longer empty.
Instead it seems to take an eternity (minutes, actually still not finished after three minutes) until the certificate cache is loaded.
Maybe the problem is the "Check Point Endpoint Security" being active on the client. It looks as if it prevents use of Kleopatra.
As I don't have administrator rights ("for security reasons"), I cannot analyze what's actually going on.

Seeing that there are "groups" in Kleopatra, I read the docs, and they suggested that the groups are for addressing multiple recipients.

Well, I think it's a matter of taste what keys a user wants to have in which keyring. Some users want only the keys they actually use.
I think policy or the algorithm should not take the decision away from the user. "Free software for unfree users" doesn't make much sense. Maybe I'm just so old that I still think the computer should do what the user wants it to do, and not the other way around (which seems to be a current trend).