Bitcoin is an amazing Payment Mechanism. But it's not a good unit of account, and possibly not a great store of value. As such, the solution to the problems posted in the parent will likely involve transmission of USD over the Bitcoin network. Banks will be the ones to do this.
When you say, "one of the large banks" I think of Chase, BofA, Wells Fargo, etc. And, my experience with these entities is that it takes 3 days for them to do ACH transfers (well, the originating account is debited ~immediately, but the destination account doesn't see the $ for 3 days). YMMV.