I don’t know enough about the big banks to make a comment or referral on those

Our “main” bank is Regions but I just looked and they are not good. They are in the southeast. We have free online banking and bill pay, images of checks, etc. All that you are looking for but I am not sure what “remote deposits” are so I don’t know if they do that.
We chose to park roughly 1/2 of our FFEF there because of hurricane evacuation. Almost anywhere we would go in the southeast has a Regions or in a nearby parish(county)/town.
Our other bank is a local bank. It is slower to implement some things but we do have free online banking through them.
These 2 banks do pretty much all we need. We have not had any purchases or transactions cancelled or rejected as you have. I am sorry to hear about your experiences.

Oh, I expect he knew exactly what he was saying

A lot of these folks count on consumers not knowing their rights and not knowing their options. They use fear to drive the reaction they want to see. The moment the consumer speaks up and says “I already know you have no legal authority to do this”, the game is over. I wouldn’t even get into the amount of the bill, how they found you after all these years, any of it. Just tell them you already know they have no legal authority to pursue payment of this bill, and if they continue to try, you’ll press charges, which DO have authority.

If the statute of limitations has really gone well beyond your state’s limits, then this is intimidation

I wonder if it would be a good strategy to call his office, inform them that this bill is well beyond the statute of limitations, and if they continue to pester you, you’ll file harassment charges. I would expect that to dry up this conversation in a hurry. Attorneys enjoy using the law to flog others but they are quite sensitive to being flogged by it when the tables are turned. And he’s clearly out of line. Use that tool; it’s there for that purpose.