Funny, the first hit for that search you linked is... my comment above. The "few things" other than that are for alternatives to PKIX, which is fine and good, but PKIX will be with us for a long time yet. As for Everest, it jives with what I wrote above, that verified implementations of TLS are feasible (Everest also implements QUIC and similar), but -surprise!- not listed is PKIX.

I know, it sounds crazy, really crazy, but PKIX is much bigger than TLS. It's big. It's just big.

The crypto, you can verify. The session and presentation layers, you can verify. Heck, maybe you can verify your app. PKIX implementations of course can be verified in principle, but in fact it would require a serious amount of resources -- it would be really expensive. I hope someone does it, to be sure.

I suppose the first step would be to come up with a small profile of PKIX that's just enough for the WebPKI. Though don't be fooled, that's not really enough because people do use "mTLS" and they do use PKINIT, and they do use IPsec (mostly just for remote access) with user certificates, and DoD has special needs and they're not the only ones. But a small profile would be a start -- a formal specification for that is within the realm of the achievable in reasonably short order, though still, it's not small.