German bank Fidor is known for sending a tx notify email immediately after each transaction with your account card.
I've been writing API code for a bunch of bitcoin-related exchanges lately, and on reflection, it seems quaint that my bank doesn't offer an API, if only to do read-only queries of transaction history and balances.
For instance, if I'm selling bitcoins on localbitcoins, and someone marks a payment sent, it'd be really nice if I could automatically verify at the bank-side, and then automatically release bitcoins to the buyer. The closest I could get to that, is if they (the bank) offered email notifications upon deposits, and I then parsed those to see if they were localbitcoins related.
So yes, I'd probably use an API, but I wouldn't expect more than read-only access for average consumer accounts.
It would beat the trouble of having to manually click to download each one, and then being presented with a suggested filename of "session.cgi" to save the thing into (i.e., they are not even smart enough to set the mime headers to give a proper filename to the file).
The problem comes in when you try to do things like "solve this for the thousand most common banks, while not making users enter their credentials every time, with adequate coordination with each bank to avoid being blocked for logging into numerous accounts, with adequate security to avoid mass-leaking online banking credentials."
Had they a real API I suppose I would use it rather than this. I have a couple of accounts for different purposes and like to avoid moving the money manually every month.