Google (and every other software that chooses to) do not have to follow 301s. It basically means "you cannot get what you want HERE. If you really want it, go THERE instead."
Notice that there are several reasons for Google's "non-reaction" to 301s, which will be apparent if you track log-files.
a) Google often has no reason to follow a 301, because it leads to a page that Google's bots have already lifted based on other links followed. In other words no reason to load it, because it is already indexed. Most cases are for this reason, in my experience. In other cases they are followed. (Google might have many other criteria.)
b) Google in my experience comes back to check old links with redirects again and again for a while, until it is sure that they are really permanent moves. In some cases it can of course be because it found the old link somewhere and hence is following it again, or because the old link is still present in an older indexed page. But in some cases I have noticed, it is merely a re-check. The latter will stop, once Google can be sure it is "really" permanent, which a 301 is supposed to be. (As opposed to a 302).
That then kills the old page off the indexes.
On the validation file checks: Since Google changed their webmaster tools to require you to register both versions (www and non-www) if you own them both, they would have to check both during re-validation of ownerships. But as per the above, if you have redirect from non-www to www, they would not have to follow the 301 redirect, since it points to a file Google would check separately (registered separately in webmaster tools).