I just tried til help a friend with installing Lightroom on Mac OS Ventura. Creative Cloud was installed, and I used the button Install on Lightroom in the Creative Cloud app. The installation was performed, and reported Completed. Lightroom button changed to Open.
Clicked on Open, but nothing happened.
Wide search in Finder for the Lightroom app file, but it was not found on the entire disk.
Edit: As Lightroom is tested compatible with Ventura, this is probably related to the installer app or lack of priveleges. See my comment below.
Regards
Arnold Hoddevik