I don't think there's that much potential for user error, it's only two settings: CC and subtitle language. Pressing the big silver button would only toggle the CC on and off, I think. I certainly understand your frustration, many times I too have come to this forum looking for help, and found lots of people who aren't having trouble. It seems you understand that CC isn't available on all shows, so that's a start (I actually called Netflix a few times to find out why my 5.1 wasn't working, only to realize that it wasn't on all shows, even very recent HD movies).
I have two thoughts on other solutions;
1. Try a factory reset. Sometimes updates get a little muddled, worth a try anyway.
2. What is your connection speed like? When I was trying to get my 5.1 working, the eager (if not helpful) people at Netflix said that if the connection speed isn't great, the software will try to reduce the requirements, including turning off the Dolby. I can't imagine that CC is such a bandwidth hog, but perhaps if your connections isn't great the AppleTV is ditching the CC to maintain image quality.
Otherwise, it's probably time to get on the phone with tech support at both Netflix and Apple. I'm sure they'll blame each other to some degree, but at least you can get the issue on their radar. There's really not too much that the user can do to troubleshoot this, so it's likely a service or device issue. Netflix is able to tell you how fast you are downloading the stream, which may be helpful (or just interesting).