I am trying to verify my keychain account, but the verification code it being sent to a phone number I don't have access to

Frequently asked questions about two-step verification for Apple ID - Apple Support

You'll need another trusted device, or your Recovery Key, which you may have elected Apple to keep for you.

I called Apple Support to accomplish this. Had the same problem and it wasn't resolvable through online methods for the exact reasons you cite. Had to answer multiple security questions for the Support Rep and then he activated the reset for me. This was only a couple of weeks back, so it definitely works!