I can't help but think that Harry dropped an incredible idiot ball on deciding to go to Azkaban. I don't mean his deciding to trust his Professor and Mentor. I'm having trouble reconciling Harry's timeline with either his or (more importantly Quirrel's) decision making style.

8 AM - "Well, I have a big day of breaking into Azkaban today. So much to set up, I've got to be super careful!"

3 PM - "Hmm, it seems that the failsafe Quirrel setup in case anyone believes I was involved in Azkaban was triggered. Better go through with the plan anyways."

5:55 PM - "Hey Mr Quirrel, our failsafe triggered and McGonagall suspects me of illegal use of my time tuner, indicating something went terribly wrong. Think we should abort the plan?"

6 PM - "Welp, time to go to Azkaban!" . . . "Oh no! It's going horribly wrong, how could we have ever anticipated this?!?"

I'm aware you can't use a time tuner to solve problems, but from Harry's/Quirrel's point of view the most logical action would be to abort the mission and send back the note to prevent paradox. I can understand that Harry has already been established as acting irrationally at this point in time. However, it is unimaginable that Quirrel, a planner far more than one level above me, would simply ignore the failsafe being triggered and continue onward.

The only option I can fathom is that Quirrel intentionally planned on failure and had them go forward with the plan anyways. But this doesn't explain why Harry doesn't use it as very strong evidence that Quirrel is evil and out to get him. Harry doesn't even reflect on the fact that, in retrospect, going onward even after they knew the failsafe was triggered was an idiotic move.

This seems be either a glaring plot hole or an idiot ball, but I may be misunderstanding all interactions of time travel or getting the timeline wrong.