Send me a 401 (or a 403) status and I’ll know I’m not authorised.
In the physical world, nobody would lawyer up and go to court if someone walked through an open door with a sign saying “public entry here” and saw something confidential.
If you have confidential information around in the physical world, you make sure you have facilities staff who know the difference between “public entry here” signs and “authorised personnel only” signs. You also have facilities staff who know how to fit door locks and door closers, and security staff who know how to choose appropriate locks and to enforce compliance of locking doors. And if all that breaks down, it’s not Joe Concerned-Citizen who tells you about it, or even Mallory from your competitor who waltzes out with trade secrets who gets held to account, it’s the manager and/or executive in charge of facilities and security who’d be answering the difficult questions, probably with their lawyer at their side.
It sad that the legal system hasn’t yet started to hold people to account for having incompetent web developers and server operators.