Sure, but it seems like all the CAs are stopping issuing certificates with the client EKU. At least LetsEncrypt and DigiCert, since by the Google requirement they can't do that and normal certs, and I guess there's not enough market to have one just for that.
> You might do that if you're using the host HTTPS certificates handed out by certbot or other CAs as mTLS client certificate
Sure, what's wrong with that?
> You can still generate your own mTLS key pairs and use them to authenticate over a connection whose hostname is verified with Let's Encrypt, which is what most people will be doing.
That lets the client verify the host, but the server doesn't know where the connection is coming from. Generating mTLS pairs means pinning and coordinated rotation and all that. Currently servers can simply keep an up to date CA store (which is common and easy), and check the subject name, freeing the client to easily rotate their cert.