That would be preferable. I had a call from my bank once that went like this.
> CS: Hello, this is X from Y Bank. Is this Z?
> ME: Yes, this is Z.
> CS: Z, can you confirm your last 4 digits of your social security number so I can confirm who I am speaking to?
> ME: Uh... How do I know you're the bank? I can't just tell any person who calls that information.
> CS: Z, we are from your bank, Y Bank. Please provide the last 4 digits of your social.
> ME: There has to be another way to do this, right?
> CS: Please hold... (puts me on hold for 30 seconds)
> CS: Hi Z, I'm back. Can you look up our phone number on Google and give us a call back and ask for me by name, X?
> ME: Sounds good.
It was them. I got quickly reconnected to the same woman. I'm still concerned that the norm should be something closer to what you commented, even if it adds friction and will probably result in lazier people not calling back for something important. It's better to normalize this than allow for the alternative, which is to normalize people telling random strangers their sensitive, personal information.
Any time this topic comes up, I immediately worry about my parents and grandparents falling for this sort of thing if real scammers are out there trying. I realize the last four digits of my social security number are not as great as the whole thing, but as far as I know, it's enough to be dangerous.