Im sorry that sucks, but we should not give up rights because some customers are easily fooled. This is a slippery slope you do not want to start on, because where does it end?
But that's more an issue of identity confirmation.
I believe there has to be a reasonably high bar that a person has to clear before a company should be even allowed to assume they are who they're claiming to be, but once that bar is cleared no information regarding or directly linked to the person in question should be withheld from them.
But then, hasn't the company already shown that it's bad at identity confirmation? Why would you expect them to be better at it in the other direction?
Isn't this problem solvable by looking at it differently? To me, the problem is that it's easy for scammers to impersonate someone. What if there was a way to reasonably check a person's identity in a standard way accessible to everyone without going through hoops?
That ideal won’t come true for a very long while. I appreciate your idealism, but for better or worse, we need a reasonable and attainable solution in the interim.