Automation for System-wide Insert of Date & Time

Yes, you can assign a keystroke to a Quick Action, though you must be careful to not use a keystroke used by the Finder or other applications. One of my Date-oriented Quick Actions uses control+option+D.

In System Preferences > Keyboard > Shortcuts > Services, look for the name of your saved Quick Action. On the right side of that entry will be the word none. Click the word none and it will change to an Add Shortcut button. Click that, and then type the control, option, and d key sequence, and then press return. The Add Shortcut button appears again. Click it to see your shortcut. Quit System Preferences.

You need to run the Quick Action once via the Application > Services menu before the unique keyboard shortcut works.

This works with Notes on macOS 11.5.2. You position your cursor in Notes where you want the date string deposited and then from Notes > Services > Date String Output. The Quick Action, in this case, was named Date String Out.

If you wanted a custom date string, and not the default AppleScript format, then you can use a Run Shell Script with the UNIX date application to create that output. Far simpler than all the AppleScript date/time string manipulations to achieve it. I can post an example if you want to see how that is done in a Quick Action.