I assume then that you exchange the public keys of your Signal contacts over SMS. Over perhaps you scp them to a server you share with friends.
> The Web PKI is something we live with because we have to, not something anyone sets out to re-create.
We have to because it's crucial. If public key attestation didn't matter we would have dispensed with the lock icon and "this certificate is untrusted" popups rather than laud the emergence of Let's Encrypt and the ACME protocol.
> I assume then that you exchange the public keys of your Signal contacts over SMS. Over perhaps you scp them to a server you share with friends.
You are meant to verify your contacts' keys (aka "safety numbers") either in-person or otherwise out-of-band. The Signal server does not in any way sign or endorse the identity keys they serve to users (yes, they're delivered over TLS but that doesn't count).
Yes, Signal does do some best-effort verification of your phone number when you register a device but that's just to avoid DoS. SMS can be easily intercepted.
There are organizations that benefit from a PKI-ier deployment of SSH, but even there, the "I" part of the PKI is extremely attenuated, and most of the real interesting work is done by a single centralized point of trust that mints time- and usage- limited token-equivalents. They're not trees so much as they are vines or fungus colonies. They're great, but they're certainly not a vindication of the 1990s concept of a PKI.