Anecdotally, I seem to have had the opposite experience. I've been doing this for at least 15 years, and never had a negative reaction. With bank, credit card, or finance-related companies, they seem to understand immediately. With other callers I've gotten awkward pauses, but ultimately they were politely accommodating or at least understanding that some issue would have to be processed through other channels or postponed.
However, I don't have strict requirements. When a simple callback to the support line on the card, bill, or invoice doesn't suffice--and more often than not it does, where any support agent can field the return call by pulling up the account notes--all I ask for at most is an extension or name that I can use when calling through a published number. I'll do all the leg work, and am actually a little more suspicious when given a specific number over the phone to then verify. Only in a few cases did I have to really dig deep into a website for a published number through which I could easily reach them. In most cases it suffices to call through a relatively well attested support number found in multiple pages or places[1].
I'm relatively confident that every American's Social Security number (not to mention DoB, home address, etc) exists in at least one black market database, so my only real practical concern is avoiding scammers who can't purchase the data at
[black] market price, which means they're not very sophisticated. A callback to a published phone number for an otherwise trusted entity that I already do business with suffices, IMO. And if I'm not already doing business with them, or if they have no legitimate reason to know something, they're not getting anything, period.
[1] I may have even once used archive.org to verify I wasn't pulling the number off a recently hacked page, as it was particularly off the beaten path and a direct line to the department--two qualities that deserve heightened scrutiny by my estimation.