I don't think that Number 2s want to extract any information from Number 6. I think that the Village is a prison in which the inmates guard themselves. They stay because their spirits are broken and they accept to act as one in spite of themselves (a metaphor for the real world). The various Number 2's objective is to break Number 6's spirit not to force him to reveal something but to force him to become yet another prisoner and act like they want him to and remain in the village all happy and all right (they need him in the village not in the outside world because he knows too much). They want him to try and escape and fail every time because they feel that is what will break him in the end. The information gambit is just to give them a "cover story" if you will. In the end, we are pretty bluntly told this by being shown that the big guard is none other than Number 6 himself i.e. the prisoners are imprisoning themselves the way we adhere to the rules and regulations of the real world through what we believe is our own free will.
Really to count, a woman must have either goodness or brains.
There's a good point in that. The ultimate aim is to make Number 6 his own willing warden and jailer, just as in today's society the ultimate goal is to equip each member with his/her own infallible, internal censor, who is much more effective than any outside agency could ever be. I think PMG very much wanted to point up the fact that every man, under the correct societal pressures, is far more a threat to his own freedom than any external force could be.
You're almost on the mark, and in fact you're very close to hitting dead center of the bullseye. I'll keep my response in Spoilers in case you really don't want to see it, because I believe Patrick McGoohan intended for viewers to make up their own minds. However, he did make some solid answers about the show in various interviews.
The show, at first glance, is all about the Individual versus the Collective, in that Number 6 is trying to keep what he thinks is private as private as he can against the wishes of the majority. But, it's really a metaphor for what's going on in one's own Mind. The Village is the Ego, the Mind's Comfort Zone, and this is controlled mainly by the Id, and not the Ego or the Super-Ego. The Village is also a collection of various knowledge, talents, skills, professions, vocations, and other things that help keep 6 sane. The Id is represented by Number 1, 2, and the apparatus that seeks to break 6 of what they want to know from him. In a sense, what we're seeing from 6 is an expression of doubts, anxieties, and other issues plaguing his concscience about his resignation. Did he do the right thing? And, if so, are his anxieties merely a reflection of what he was brought up to believe by the society he was raised in?
Number 6, really, is the Super-Ego here. And sometimes, if you note, he gets the better of the Id during all this, and finally triumphs thanks to his Will, guided by discipline and a rather distinct antipathy for the Id itself, the monsters that dwell down there that everyone has. One thing I think that The Prisoner did not really address too fully was what would happen if The Village, and the Id, were something he appreciated, and understood, and liked, would he be so antagonistic to them? When Number 2 tries to subvert 6 into compliance with his or her schemes, this is mainly a subversion of the Super-Ego, not an attempt really to change in accordance to 6's liking. And they wouldn't anyway, because Patrick McGoohan himself suggested that The Village and the apparatus operating it all would be the least likely to change, and indeed it really doesn't at the end.
To punctuate this, I mentioned in another thread the episode Many Happy Returns, where 6 wakes up only to find the entire Village deserted. They up and just disappeared, all except a lingering black cat. Now, he knows, by now, what this place is capable of, and he's made escape attempts before, only to be betrayed by people he personally knew in the Service and sent right back to the Village. Here, he escapes by building a raft and decides to go to, of all places, England, where he visits his own flat now owned by someone else, and decides to go find help from his old pals again! Why does he do that? You or I, in that situation, probably would never have gone directly home much less back to the very people who put us in that situation, trying our chances on the continent or maybe even going to America or even, dare I say, to the enemy, for help. The last thing we'd probably do is attempt to find the Village again as he does, and that's how he gets right back there again. He lets his guard down, finds his comfort zone, and that comfort zone betrays him, as it always will.
But, in the end, Number 6 is still a Prisoner. Freedom is an illusion, according to McGoohan, in the sense that we're all prisoners of the things in our mind that keep us sane, comfortable, and operative. We may never know fully why he resigned, only that it was a matter of conscience. And it may haunt him for the rest of his life.