A client cert can be stored, so it provides at least a little bit of identification certainty. It's very hard to steal or impersonate a specific client cert, so the site has a high likelihood of knowing you're the same person you were when you connected before (even though the initial connection may very well not have ID'd the correct person!). That has value.
replyBut it also doesn't involve any particular trust in the CA either. Lets Encrypt has nothing to offer here so there's no reason for them to try to make promises.
Eh, it's pretty easy to impersonate if the values in the certificate aren't checked, and you could get one from any of a list of public CAs.
replyIf you're relying on a certificate for authentication - issue it yourself.
Point being that if you get a valid TLS connection from a client cert, and then you get another valid connection from the same cert tomorrow, you can be very certain that the entity connecting is either the same software environment that connected earlier, or an attacker that has compromised it. You can be cryptographically certain that it is not an attacker that hasn't effected a full compromise of your client.
replyAnd there's value there, if you're a server. It's why XMPP wants federated servers to authenticate themselves with certificates in the first place.