The answer is float. When you send a bill pay payment your bank debits your account immediately and transfers the funds into their temporary holding account.
They then issue their check to the payee and mail it out. A day or two later. Three or four days after that your payment is received and posted. In the meantime your bank collects interest on your money that they have held for three or four days. Who cares, right? Maybe less than a penny in interest, but when you figure this transaction times millions each year, It makes money for the bank. The bank wins and you win by not having to spend $.50 for a stamp.
Well, back to why the bank needs the post office, they don’t. If I get a check I have an app I can photo the check and deposit instantly. Why can’t a bank do the same to another bank? You answered it. They don’t want to. That’s no reason to keep mail service.