Serious question: Why would any financial institution implement something that will reduce their revenues?
Followup question: How would a company be able to come in and offer what you describe and offer users savings, and take a cut for themselves along the way, so that the users and the new company wins, while the incumbent banks lose?
Some banks offer something a little bit in this direction (at least in the UK and Australia). I have heard of banks with 'savings pots' and online saving planners.
For a bank it is probably good for them if you save. Eventually you will spend the money and in the mean time the saved money makes them money because (due to how the banking system works).
The way I can see this playing out is based on consumer demand. Banks have little in the way of genuine USP. I mean they all pretty much offer the same service and must compete on service, interest rates, fees, branding, reputation etc. So offering a way to help people save could be a good USP for the right kind of bank.
Followup question: How would a company be able to come in and offer what you describe and offer users savings, and take a cut for themselves along the way, so that the users and the new company wins, while the incumbent banks lose?