There are new entrants to the market such as Felix1 in Germany or Crunch in the UK though that promise this exact service.
Your suggestion certainly is a good one but I'm still wondering why accounting and book keeping requires so much human-machine interaction. I'm not talking about handling edge cases, exceptions or conflict resolution, things you need a human being for. You shouldn't need an accounting firm for entering incoming invoices or synchronizing your bank account statements.
Also, a human should advise and help you to move to another bank, one that has better interfaces with modern software solutions (like the one you use or Quickbooks or the many other around).
I tell you from my experience working in finance that you will never manage bank reconciliations automatically 100%.
Famous last words. As long as you define 100% to be residual errors at human level.