I retested with the old Mouse Pick Event. It works so far. If there's any problems, I'll come back here.
Otherwise, sorry to have bothered you :-)
Edit:
Everything is working fine now. I don't know what happened. It was a large project that had been migrated, and something was interfering with the click functionality.
Sometimes a click doesn't work, but if I exit and then re-enter the scene (build), it works.