To which the response will inevitably be "Communication with the bank can't be trusted" and many people will read as "The bank can't be trusted". I think people will quickly come to the conclusion that the stakes are much higher for them if the bank can't be trusted when they have their money there, compared to the browser being too assertive, and will just move their money. Without definitive knowledge or a good understanding of all the intricacies, that's the safe decision.

Banks know this. Like the CA system, the whole banking system only functions because of trust, and in the US that trust is backed by the government. They aren't going to let that erode. Regardless of whether all their back-end certs get updated, their customer facing ones will if needed.