I think there’s two things going on here. The first is that Harry is psychologically in fantasy-mode during these chapters, and the second is Harry’s self-esteem issues regarding his own intelligence.
“You are about to invite me to join a secret organization full of interesting people like yourself,” said Harry, “one of whose goals is to reform or overthrow the government of magical Britain, and yes, I’m in.”
Fantasy-mode: Harry is being recruited by a secret group of highly interesting rebels. They fight against the stupid, evil, corrupt government of Magical Britain. Their cause and methods are righteous beyond question (otherwise Harry would ask a few, instead of immediately inducting himself).
Quirrell believes Magical Britain must be ruled under the dictatorship of a powerful leader, as we learned in chapters 34-35 (whereas Harry believes in democracy). So what kind of secret rebel organization is he likely to be a member of? It doesn’t matter. Harry is in fantasy-mode—he could be in a secret organization of interesting people whose goal it is to change the world!
Fantasy-mode is completely obvious throughout these chapters, especially at the start of 52.
Self-esteem issues: This thing with Quirrell being able to make “amazing deductions from scanty evidence” has been brought up before. Quirrell has also told him things like “You should have figured this out”, “you’re childish”, and sometimes it even seems like Quirrell is testing Harry’s intelligence. This is making Harry insecure, and even desperate now. He’s thinking a week in advance of how he’ll answer Quirrell’s questions, rather than suffer the humility of having not already deduced and fully understood the secret plan by the time he was asked.
Quirrell is playing Harry on at least these levels: “save the world” fantasy, and “you are not as intelligent as I” subtle cues. These weaknesses of Harry are apparent in a lot of previous chapters. Since these are established vulnerabilities, it’s plausible that Quirrell can successfully exploit them without Harry knowing.
The business with Snake-Quirrell whispering instructions to Harry might suggest the Imperius Curse. In Rowling’s book #4, Moody casts the curse on students and that’s just what it’s like—verbal commands that are followed without question (unless you’re trained in resisting the curse). Bellatrix doesn’t seem to notice that Harrymort is talking to his snake. Perhaps Voldemort was known to do this all the time, but it could be because the instructions were being issued directly to Harry’s brain.
But the fact that they can’t cast magic on each other is a big obstacle for this theory. Of course, a key point in these chapters is that it’s possible to control somebody without ever Imperiusing them.