OK, if you define the availability of writes in this way, that they must complete immediately, then indeed the availability part of CAP is not met.
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).
Yes, but if you're allowed to timeout, then any system is trivially available, even a system where the nodes are always offline (simply timeout every request).
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).