If you can count on a prior relationship in encryption, then you don't even really need PKI. Just mutually decide on an encryption key and then use that to encrypt/decrypt your traffic.
Not every website is Twitter or Facebook or Google or a major bank, where the vast majority of visits are repeat visitors. I run dozens of websites, and every single one has more than 50% new visitors, year in and year out. That is representative of the vast majority of websites out there.
Browser makers have dramatically shifted their security focus toward repeat visitors. Two-factor authentication, same-origin policy, HSTS, and other new security features are only useful for repeat visitors.
EV certificate notices (the "green banner" in the address bar) are one of the very few browser features that attempt to help first-time site visitors. Now, obviously EV certs have problems as you cite with Stripe... but instead of trying to solve this problem, browser makers are simply copping out and falling back on a posture that HTTPS just means encrypted over the wire, not identity. Essentially: establishing security is the user's problem. Once it's established, then we've got all sorts of tools to maintain it.
This is a mistake. It does not help most websites, and it privileges incumbent services (which have lots of repeat visitors) over new websites (who are trying to win new customers).
CAs and browser makers must continue to attack the problem of trust on first visit. It's not an easy problem but it is the essence of the promise of the web. Visitors should be able to load a new site and know if it is trustworthy, just like people trust that they can physically walk into a new store and not get robbed.