I've tried it now on 22H2 and got some interesting results, and a possible workround for you.
I can reliably replicate what you are seeing, and I can get round it quite easily.
The testing I did:
I created a new standard user and installed CCleaner Free from that user account (admin password req'd of course).
I didn't open it from the installer when it asked.
After installation CCleaner wouldn't open.
Uninstalled (and cleaned up using Revo).
Installed CCleaner again as the standard user, (admin password req'd), but this time I opened it as the last step when the installer asked.
It opened from the installer and ran normally, I closed CCleaner.
It would then open as normal for the standard user from the desktop icon, and also from the exe in File Explorer.
Uninstalled (and cleaned up using Revo).
To test it a bit more:
I signed out of the standard account, logged back in as Admin and installed CCleaner from there, I didn't run it from the installer.
Logged out, and back into the standard account.
CCleaner would not open for the standard user.
Back to the Admin account, (and a quick double check that CCleaner ran there as normal, it did).
Uninstalled (and cleaned up using Revo).
Installed from the admin account again but this time I ran CCleaner from the installer when it asked.
Logged out and back into the standard account.
CCleaner would open from the desktop and from File Explorer as normal for the standard user.
So the solution appears to be that if you want a standard user to be able to use CCleaner then you now have to run it from the installer first time when installing.
If you don't do that then it won't run for a non-admin user.
I've not tried to replicate it again on 21H2 but I assume it will be the same.
It appears that running CCleaner initially from within the installer is setting something 'extra' that doesn't get set if you don't run it that one time from the installer.
(And of course you always have admin permissions to use the installer, so also for that first run of CCleaner from it).
Why it should be doing that I have no idea, from your own observations it must be tied to the folder permissions somehow.
I'll point it to the staff for the attention of the developers.
Of course an easy fix would seem to be having it <em>always</em> run from the installer rather than giving the option not to. Many other software installers do that as standard and alway open the newly installed app.
Please let us know if runing CCleaner initially from the installer, also cures the issue for you like it does for me.
(I've thought of another couple of things to try/test later, but that's more for my own interest and shouldn't add much to the above if anything).