Yes, with Postfix you have to enable that it uses them in the config. I think that is smtpd_tls_ask_ccert = yes plus smtpd_tls_received_header = yes.
In sendmail, however, this is the default, and it actively displays verification FAIL in the Received header if the certificate of the sending MTA isn’t qualified as TLS client (which it is, in this setting, even if it by itself is a server).
This can be generalised to all machine–machine communication (whereas the cited use of webbrowsers is generally human–machine only)… incidentally also for HTTPS where API use is secured by SSL certificates and the endpoints are both servers (and as such both qualify for an LE certificate with domain validation). One setting I’ve actually set up is using a smarthost for outgoing eMail, and the smarthost verifies the sender with certificates (as SASL is much more expensive and increases the attack surface).