I think the message is probably incorrect in the fact they haven’t interviewed too many candidates. I wonder if you got a “template” rejection that was sent to people at all stages of the recruitment process?
I wouldn’t ask on LinkedIn. I would go back to the recruiter and ask the feedback given you have been interviewed and basically insinuate that you will consider raising a Subject Access Request to obtain as much detail as possible if they aren’t able to provide a short summary as to why you have been unsuccessful.
The only time I wouldn’t do this is if it was a pre-recorded questions video interview. Then I wouldn’t hold out much hope for feedback given the higher volumes of candidates who will complete that stage.