I'm clearly old and stupid. I can't figure out how to submit an issue for #GnuTLS on their gitlab anymore. (I did in the past.)
So here it comes instead: gnutls_certificate_verify_peers2() does not seem to verify ExtendedKeyUsage but gnutls_certificate_verify_peers() does.
Neither case is documented clearly. This has already lead to people submitting vuln reports to gnutls-using apps for this omission.