As of yesterday, I can no longer start GOG Galaxy. Windows gives the error message that it "cannot find <install location>\GOG Galaxy\GalaxyClient.exe".

I've tried uninstalling, reinstalling the newest version and restarting PC; it still gives the same error. Does anyone know what's causing this or how it can be fixed?

Have you tried launching the executable directly from the installation location? The default is
C:\Program Files (x86)\GOG Galaxy\GalaxyClient.exe

You may have outdated shortcuts or your antivirus may be placing files in quarantine.

Yes

May be too obvious but is the File there? Is the general directory existing?
If the directory exists but the .exe is missing i would guess on antivirus program that got hungry.
If the whole directory is missing either the install location is wrong or the install got blocked/went totally wrong.
In that case i'd start a search for 'Galaxy' over all drives, maybe it's just somewhere else and the link is outdated.

If there is no trace to be found try to install it again and either open the location in the explorer to see if something gets written there or as an advanced possibility get the microsoft sysinterals suite, start procmon and watch with that what is happening.