Earlier quoted context omitted.
Robots.txt is not a 'no-trespassing' sign. Robots.txt is a 'whites-only' sign. The information is available to the public, just not for certain classes. This is and should be legally unenforceable. If something is truly meant to be private it should not be referenced from a public-facing page or it should have access control enabled.
Except race /skin color is a legally protected class, and robots aren't (and why should they be? They can't enter into contracts, conduct business, etc. So it's perfectly legitimate to exclude them from a site where they cannot use it in the intended manner). "If you truly didn't want trespassers you should've put up a gate."
As a property owner, a no-trespassing sign won't protect you from the lawsuits that result when a toddler drowns in your pool. You're expected to do more (like putting up that gate).
Equifax's systems are peppered with "no-trespassing" motds at login. They also have a robots.txt file. We expected them to do more.
Same for leaving keys in your ignition, guns unlocked on your nightstand, etc. "Don't touch" signs won't absolve you of responsibility when either gets stolen and used in a spree killing.
So yes, as the owner of any sort of asset, in most contexts it is your responsibility to implement access controls to keep unauthorized traffic out.