The alarming thing about being tested isn’t that they tested Harry specifically; it is that they were aware that tests needed to be performed at all. Had the plan gone successfully, nobody would have ever have known that Bellatrix was removed. Remember that the entire advantage of Patronus 2.0 (to Quirrell) was the undetectable nature of it. It allowed a person to commit a perfect crime without the guards or dementors being aware that a crime ever occurred.
From the point of view of Harry, sitting with his invisibility cloak in the empty room at 3:00 PM, only a small number of possible futures could exist:
No note—Either the plan will go successfully, Harry shall be captured/killed in the attempt, or they abort for some other reason.
Do not mess with time note—This also should result in aborting the mission since some terrible paradox occurred and you DO NOT mess with time.
Passcode note—Harry will not be apprehended, but the plan will fail in some way and Harry is a suspect. Or Harry will chose to abort the mission and send back a false passcode preserve a stable time-loop. This is the result that actually occurred and is the only one that confirms a definite partial failure.
I admit it is possible that, once the message was sent back in time, Harry and Quirrell were committed via fate to perform the prison break. The problem is that both Harry and Quirrell act as if No Note was received. In the TPSE chapters, we do not see characters who are aware that their plan will be detected. They do not seem to act as if their plan is definitely going to partially fail. Neither does Harry take heart in nor mention the fact that he has already survived escaping Azkaban. Contrast with the same situation in cannon!PrisonerofAzkaban where it is a major plot point.
Edit: There is the possibility it is related to “Azkaban’s future cannot interact with it’s past”, but then you run into the problem of Harry being able to send the note at all. If they abort or are just more paranoid on their mission, Azkaban’s future is still effecting its the past either way.
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.