What should Luke have done, hire a private investigator? I think a Jedi Master has the right to investigate a student suspected of flirting with the dark side. You can’t apply real world concepts of privacy to the GFFA.
Maybe if the Jedi council had been more inquisitive about what Anakin did on Tatooine, or where he often spent his nights, the galaxy would be a different place.
Well it’s all about trust, as Rey notes in the film, I believe. TLJ makes a strong case, that Luke’s actions pushed Ben over the edge, resulting in Ben destroying the Jedi Temple over Luke’s betrayal. The problem is that TLJ doesn’t bother to explain what put Luke in this state of mind (or Ben for that matter) other than sensing some darkness. That apparently is enough to get the more reasonable options off the table, and sneak into your students room to probe him for information. He might have confronted Ben, if he felt uneasy about the situation, rather than trying to obtain the information in such a dodgy way. In my view there needed to have been some sort of incident in the film, that caused Luke to lose his cool, and cross some boundaries in order to protect his other students.