Next step is to check whether the shortcuts are being recorded in the preferences when LayOut quits. Set some shortcuts and then quit LayOut. The shortcuts should then be contained in folder
~/Library/Application Support/SketchUp 2022/Layout/
in the file layout.private.json
Your user ~/Library is normally hidden by Finder (it is NOT the same as the global /Library folder!). You can get there by the Finder’s Go->Go to Folder menu item by typing the path into its dialog (make sure to include the right capitalization and spaces). The ~ is just traditional shorthand for /Users/yourusername/ folder, where yourusername is replaced by your macOs login name. You can also type the full string explicitly.
First look at the json file’s info (right click on it in Finder and choose Get Info) and see whether it was updated the last time you ran LayOut. If not, there is a problem writing to that file, most likely with folder or file permissions, so check them in that same info panel.
If the file was updated, view it with any text editor (or just select it and tap the space bar to get a text view). Scroll down until you find a section labeled “shortcuts.dictionary”. This contains a list of all the shortcuts LayOut knows about. Each shortcut gets three lines akin to:
The action string should be similar to the LayOut command you set a shortcut for (the exact strings differ between the Preferences Shortcuts panel and the internal form shown in the json, but they correspond closely enough that you should be able to understand them). The key and modifier values will be the Mac internal codes for whatever keychord you set for the shortcut - don’t worry about the specific values, at this point you just want to make sure they are there.
If those lines are not present in the json, there is a problem writing to the file as LayOut quits. If they are present, there is a problem reading the file as LayOut launches. Either of these is most likely the result of file or folder permissions.
Edit: one further thought - if you are trying to set shortcuts that include the command key, they won’t stick because they are preempted by macOS and cleared every time LayOut launches.