OK, so let's see how this applies to Obama specifically. The argument, so far as I understand it, goes like this.
1. Before Obama was president, his phone was tapped.
2. He must have said, or been party to, some incriminating things, because that happens to basically everyone.
3. Therefore, he could be blackmailed.
4. Therefore, he has been blackmailed.
5. Therefore, he is now doing exactly what whoever blackmailed him wants.
I have no trouble believing #1.
#2 is doubtless true for some senses of "incriminating". I think he admitted (it's ridiculous for this to be an admission, but never mind) that he smoked some cannabis as a student. He went to a church whose leader said some kinda dumb things. He must surely have said some kinda dumb things. But ... the subset of these "incriminating" things that got out during his election campaign wasn't enough to lose him the primary or the presidential election. There'd need to be something quite a lot worse to provide enough blackmail material to (e.g.) kick him out of the job or put him in prison. Richlieu notwithstanding, six randomly chosen lines do not generally suffice to hang someone in the present-day USA.
Now, of course it's possible that he really did do something bad enough to provide real blackmail leverage. But I wouldn't want to bet on it.
#3 might follow if #2 were right for a sufficiently serious sense of "incriminating". But (see above) I don't think it is.
#4 wouldn't follow from #3 in any case. It does, astonishingly, sometimes happen that people who could be blackmailed aren't. Blackmail is a high-risk strategy; if you try it and your victim isn't playing, it can get awfully embarrassing for the would-be blackmailer. And it's not as if the President of the United States of America lacks the resources to make things difficult for someone trying to blackmail him.
(He might lack the resources if he were somehow being blackmailed by the entirety of his staff, or an all-powerful conspiracy that could frustrate everything he might try to do to expose them. But that would require a size and scale of conspiracy that (a) is improbable a priori, (b) is more improbable given that no insider has ever felt guilty enough to blow it open, and (c) hasn't had all the people claiming such conspiracies mysteriously silenced.)
#5 doesn't follow from #4. Push a blackmailee too hard and they'll stop cooperating.
So: I still don't see how "is a puppet" is supposed to follow from "has been under surveillance". Too many gaps.
I assume the worry is they'll do what J. Edgar Hoover did and amass secret dossiers they can use for blackmail (or to tip off the press to get rid of) their political opponents.
Even if the president himself is a saint with nothing to hide or blackmail him over, he has many friends and allies.
With that said, any control would have to be limited enough that those controlled went along with it instead of fighting it - a relationship of complete puppetry would be difficult, as they might decide to go down and take you with them.
