I don't agree - availability is a totally meaningless property if you are allowed to occasionally return "no, I won't process your request". Such a response communicates nothing about the state of the atomic object you are writing to or reading from, so you can always return it and trivially satisfy 'availability' if we define it this way.
Availability of the individual node, or of the service as a whole? So long as the nodes always answer quickly, can't I just ask a few different ones until I get a successful response (or conclude that the entire system is down)?
The system may be unable to give you a consistent response, no matter who you ask. It really depends on how you build your protocol.
Let's imagine a system where you want to be 100% available for reads, for any number of failures less than N. Then you need to be able to submit every single write to every single node in the system, otherwise the failure of all but the up-to-date node will result in stale reads.
But then if a single node is partitioned from the network, we can't (correctly) be available for writes, because the system is incapable of sending updates to all reads as required. It doesn't matter which node you ask.
The point is that every system has a failure mode like this. I take your point that it's not always just a single node failure that precipitates the abandonment of C or A, but that was never the point of the CAP theorem.