There are posts today about NA 3.22.1 also hanging on MacOS Tahoe. (I'm about to try that).
It self healed today. Suspect something was changed server side that fixed it, since I didn't have time to go through this procedure: https://support.native-instruments.com/hc/en-us/articles/18832455607069-Native-Access-freezes-on-Loading-Products-at-startup