Why was Hagrid sent to Azkaban?

Another factor is the justice system in the Wizarding world is very...primitive. By which I mean there seems to be no actual mechanism for justice, as we know it.

Sounds harsh, but there's evidence. I doubt Hagrid was formally charged, he was expelled instead of arrested, and the business of the school and the courts was established as separate Harry's trial in OOTP. Getting questioned in CoS might make sense, even arrested - but he was not arrested, he was "disappeared"...no evidence, no trial, no appeal, and "nothing more said" if he should happen to be proved innocent (without an investigation, mind). Nevermind the isolation and torture, nobody objected, right?

The case of Sirius Black is fairly well chewed over in the book - that he was still going to be kissed after having the kids tell Fudge is Fudge's fault. The fact he was denied a trial for several years more is a lot more troublesome, because even people (Dumbledore, Tonks, Shacklebolt) being able to find evidence he hadn't any due process of law wasn't enough to get the kill order even delayed enough for trial or investigation or even just questioning.

Barty Crouch is the other case that really shows the justice system for all its flaws. His trial was short, "an excuse for crouch to show how much he hated the boy", with three people who proudly plead guilty and one who got dragged from the courtroom screaming how he didn't do it. Granted, readers know he's guilty - the question is who believed it when. there were accusations, and there was the question of if the crimes were enough for Azkaban - but not if they thought he was guilty. (Bonus point, that might have influenced his mother to get him out). And, after he's found at the end of GoF, Dumbledore questions him (unofficially, no paperwork). Fudge murders him - well, has him dementor-kissed, no questioning, and once again no trial.

In any case, the relevant term here is "summary execution" - to be accused is to be guilty is to be killed. Or tortured, in Hagrid's case. No trial, or a show trial, no 'due process of law" or equivalent thereof.

The justice system in the wizarding world seems not to run on a system of "rule of law". The Wizengamoot as a whole votes on accusations of crime in the GoF flashbacks, not a disinterested jury or judge, and there seem to be no lawyers. Instead, what we see a lot of is the question of personal integrity and reputation - Dumbledore's reputation protects Snape, again without trial, questioning, proof; Bagman is judged very lightly because of support from sports fans, rather than evidence he was fooled; Fudge defends the DE's t Harry's accusation because they are respectable, and have "good families", and donations to good causes.

I am willing to set aside the lack of anyone investigating anything and the general ability to disregard rules and laws (including the DEs) as being plot necessities... but the court system described does seem to be operating on an older basis, a rule of men, which might be on purpose since the wizarding world seems to be traditional and historical in some of its views. I suppose a people that age slowly would tend to be more conservative.