Actually, it is the opposite.
You seem to be unaware of the fact that servers do receive certificates from the clients which are then parsed.
Which is already mentioned in the advisory document:
"Thus vulnerable situations include:
- TLS clients consuming server certificates
- TLS servers consuming client certificates <---- here
- Hosting providers taking certificates or private keys from customers
- Certificate authorities parsing certification requests from subscribers
- Anything else which parses ASN.1 elliptic curve parameters"