However his definition reads: "every request received by a non-failing node in the system must result in a response"
It doesn't say anything about the timing of response. The network partition will be resolved eventually and thus the write operation will complete. Or it can time out and return an error, which is also fine based on the definition of availability (must result in a response).
The way I saw the proof go is: to satisfy CAP, you need replication, and you need high availability. To be consistent you need replication. But if you don’t have a reliable network, you don’t get reliable replication. Therefore you can’t be consistent. To me, this is a trivial result, and I’m not sure that this is what the CAP theorem intends to say?
It would be better to do, as you say, assume that a system is CA, And prove why it can’t be partition tolerant. Saying “network is unreliable” isn’t a suitable jump in argument IMO.