Firstly, nobody is actually calling for authentication using client certificates. We use "normal" server certificates and validate the usual way
I'm not sure I understand this point. You authenticate the data you receive using the client's certificate. How is that "nobody is calling for authentication using client certificates"? Maybe there's some nuance I'm missing here but if you're authenticating the data you're receiving based on the client's certificate, then how is that "validating the usual way"?
There is a lot of confusion caused by overlapping terminology in this issue.
By "client certificates" I mean (and generally take most others in this thread to mean) certificates which have been issues with the clientAuth key purpose defined in RFC 5280. This is the key purpose that Let's Encrypt will no longer be including in their certificates, and what this whole change is about.
However when one server connects to another server, all of TCP, TLS and the application code see the initiating party as a "client", which is distinct from say, an "XMPP client" which is an end-user application running on e.g. some laptop or phone.
The comment I was responding to clearly specified " I don't see how TLS-with-client-EKU [...]" which was more specific, however I used the more vague term "client certificates" to refer to the same thing in my response for brevity (thinking it would be clear from the context). Hope that clarifies things!