To be honest, I am no expert on Matrix, but I thought you can just host your own server on your own domain. That was the base assumption for my comment and I hope that is correct. They have a server reference component (synapse) that you could host anywhere I think.

Right but maybe you don't want to host your own server, but do want your own domain. Like emails.

Element's primary paid service is Element Matrix Services [1] which is exactly for that. They run a Matrix server for you, optionally on a domain that you provide.

[1]: https://ems.element.io/

That's not possible yet, see this proposal

https://github.com/matrix-org/matrix-doc/blob/neilalexander/...

The proposal that you linked is all about portable identities, tied to public keys. Is that really necessary for Element to host a server on a user's domain?

I would have thought it would work just like outsourcing your web server or your email service. You generate a DNS name and point it at their IP address. Then they get a certificate and start serving traffic for your domain. Easy peasy.

That's not possible yet, see

https://github.com/matrix-org/matrix-doc/blob/neilalexander/...

Yeah, they've had that for a long time. I ran my own server on my own domain name a couple years ago or so.