This is very simple, really: Any browser that implements this, I will not use. So any webpage that requires that API to be present, I will not be able to use. If your webpage requires this, I will not be a user of your website. It is really that simple.
Sounds great in theory but I'd suspect that you'd cave pretty soon after your bank adopts this (or whatever essential site/service you aren't considering is captured here).
There are a lot of banks, and jumping ships and reopening my account elsewhere is always a choice.