Though, mind you that "local app" is not the only way to run software. I could also install software on my home server to, I dunno, send me XMPP messages for incoming transactions, dunno whether you would call that a "local app"?
The objection to software you manage is the same as the objection to software you write - that you won't have the checks, balances, and liabilities of registered suppliers and this would create uninsurable risk that the banking ecosystem can't be expected to bear.
Though, mind you that "local app" is not the only way to run software. I could also install software on my home server to, I dunno, send me XMPP messages for incoming transactions, dunno whether you would call that a "local app"?