I'm going to disagree with some of my colleagues and suggest that the Bank's solution is much safer than your proposed alternative. Though it's not ideal either. Let's look at each solution and it's attack profile.
Address, SSN, DOB, etc.
We should all be able to agree that these are absolutely terrible ideas. Facts make horrible passwords and should never be used. No exceptions. Literally no exceptions. If someone can research it, then you can't use it to authenticate yourself. Period. Nearly all security questions fall flat on this rule. It's embarrassing. This technique is exploited frequently, often with high-profile targets. We should know better but we keep using it. This needs to stop.
While not ideal, this solves the research problem. It can still be used with secure hashing since the support rep should have to type in the password verbatim to verify it rather than read it off the screen. Your potential attacks are: (a) someone listening can re-use it to impersonate you over the phone, and (b) someone listening can use it on the website. Still, if more companies did this we'd be a lot safer. We can do better, but at least you don't have to worry about someone looking up what street you grew up on or the last digits of your credit card number and then using that information to take over your account.
This is Godaddy's solution. You have a 4-digit number on your account that you must provide when you call support. It's better as it solves problem (b) above -- you can't use the call-in pin to log in to the website. But problem (a) remains. The pin doesn't change unless you reset it, so an attacker who hears your conversation can impersonate you. Besides the below, other "call in passwords" tend to fall into this category and are susceptible to "replay attacks".
This is PayPal's solution, and it's the best solution available. When you click "contact us" on your PayPal account, you're given a single-use time-limited 6-digit pass-code that you must provide over the phone. It expires after 60 minutes or after you use it once, so an eavesdropper can't impersonate you. And it's random so it can't be predicted. If you're going to implement a solution for yourself, this is the way to do it. PayPal has been doing this for many years and it's worked for them so far.
The "tell me some facts about yourself" way of authenticating over the phone, as you suggested, really shouldn't be allowed; it's a shame so many companies do this. Giving your website password over the phone is much better (to be fair, anything is better), but it presents some problems. Authenticating online and then getting a one-time secret code -- that's a solution I can respect.
Note that the fact that the bank is willing to fall-back to asking for your SSN or Address, etc., is the real scandal here. If you refuse to authenticate yourself securely, you should not be given the option to provide some facts about the victim you're trying to impersonate.