Afaik, the LE team is distributed across the globe.
> If somebody took them out all at once, would the web's security essentially crumble?
No, there are other both free and paid CAs
> We have shitty hacks, like "serve this unique file on this web server that this domain record is pointing to", or "answer an e-mail on one of 20 addresses at this domain", etc.
Yes, but we also have certificate transparency. You can monitor all certificates issued to your domains and revoke them if needed. Not perfect but imo still reasonably safe considering you know that all the issued certs are on your servers.
> You tie a private key to domain ownership, and a private key to a domain record. Then you only have to trust registrars' keys/certs, and you can walk backward along a cryptographically-signed web of trust.
That exists and is called DNSSEC. If you haven't heard of it, you already understand: it isn't widely used. Also, it would require major rethinking of how we use the internet. Most clients do not validate DNSSEC, only public and maybe ISP resolvers do, but they can (and probably will) tamper the DNSSEC answers if they can better spy and mitm you.
> Your browser trusts the registrar's key X
Sure, we could do it in browsers, but the internet is wider than the web, and we would need to rewrite a great part or what we use every day (not saying that we can't or should not).
In the mean time, if you use a DNSSEC-compatible TLD and registrar, you can already sign your zones. That way, the current CAs will be able to cryptographically verify that the server asking for a cert also owns the domain/subdomain.