You are saving to the VirtualStore which in Windows is virtually mapped and pretends to be C:\Program Files when you aren’t running the program as admin (this is a Window security feature and working as intended). and saving files in C:\Program Files and C:\Program Data. So inside the program, you’ll see a fake C:\Program Files among other things but outside in explorer you won’t find it in that exact spot. But, please don’t save work items in those folders, its generally bad practice, and its best to use your user folder, i.e. Documents, Desktop or even create a new folder in your user root C:\Users\%USERNAME%\.
To find your files and move them somewhere else go into
C:\Users%USERNAME%\AppData\Local\VirtualStore then go into the correct Program Files (x86) or non 86. Find the kicad folder. Take the data out and delete the kicad folder in virtual store so that it stops mapping it in the file dialogs.
And if you somehow got unlucky enough with permission muckery, there may even be a C:\Users\%USERNAME%\AppData\Local\Users\%USERNAME%\AppData\Local\KiCad
which would explain the fp-lib-table behavior you see.
You will be forced to escalate to admin to run the installer regardless. Program Files always requires admin to write to and KiCad does not do user level installs (unfortunately there are a few issues in the program and not just installer that prevent it)