Ha! Or maybe Eliezer has been rolling his eyes at us (or, rather, y'all), and gave us a blatant hint with the contrast of competent Quirrell interrogating sneaky Snape and less experienced H&C working on naive Hermione. I think you're just clinging to your one beautiful idea, instead of examining other possibilities - like, say, H&C is taking instructions from Quirrell, maybe?

See? Two can play that game.