Why Did the Ministry Allow Hagrid to Stay on the Hogwarts Grounds?

Question

We now know who was the real Heir of Slytherin (spoiler: it was Tom Riddle). But back in 1968, when the chamber opened for the first time, all believed Hagrid was to blame, to the point of wand snapping and expulsion. We know the Ministry of Magic was aware of Hagrid's "guilt", because they were on the verge of shutting Hogwarts down, which didn't end up happening:

As a matter of fact, the Ministry of Magic is even now talking about closing the school.

If Headmaster Dippet and the Ministry were sure Hagrid was the culprit, why would he be allowed to remain on the grounds, endangering the lives of all the children? Furthermore, it is more than safe to assume the parents of the Hogwarts students were aware of the situation. How could they have agreed to send their children back knowing Hagrid was still there, especially the muggle-borns and half-bloods?

Even if we assume the Ministry believed Hagrid couldn't open the chamber without his wand, they still thought he was a murderer! As powerful as we know Dumbledore to be, he was just a teacher at the time, and as we've learned when Umbridge fired Trelawney, only the headmaster has the authority to determine who stays and who leaves the grounds.

Answer 1

Dumbledore persuaded Dippet. That's really all we know on the subject

‘Only the Transfiguration teacher, Dumbledore, seemed to think Hagrid was innocent. He persuaded Dippet to keep Hagrid and train him as gamekeeper. Yes, I think Dumbledore might have guessed. Dumbledore never seemed to like me as much as the other teachers did …’

_{^{HP: Chamber of Secrets - Chapter 19}}

That being said, given that he was expelled solely on the word of a prefect, if Hagrid had decided to go to the press or had been interviewed by Hogwart's board of governors, they would presumably have instantly identified that he had neither the wit, nor the ability to have committed the attacks:

The whole thing reeks of a cover-up. Presumably Dumbledore pointed out that it was better to have Hagrid on the staff and keeping his mouth shut then talking to who-knows-who and messing things up.

Answer 2

Following on from Richard's answer ...

From your question:

Even if we assume the Ministry believed Hagrid couldn't open the chamber without his wand,

While the exact location of the Chamber was unknown, it was known to be within Hogwarts proper. If Hagrid was kept under supervision, and not allowed inside the school, and not allowed a wand, the risk would be minimal. Under the circumstances it isn't surprising that Dumbledore was able to persuade Dippet of that much; he was a persuasive man, and at that time universally well-trusted.

(No doubt you felt it would be unsafe to argue against him? After all, you already had what you wanted: Hogwarts would remain open.)

they still thought he was a murderer!

No, I don't think they did. Remember, you didn't try to push things that far. Your words to Hagrid were:

'I don't think you meant to kill anyone. But monsters don't make good pets. I suppose you just let it out for exercise and - '

In other words, it was considered an accident. Hagrid finding the Chamber was unlikely enough, and as you no doubt realized, Hagrid as deliberate murderer would never have flown.

How could they have agreed to send their children back knowing Hagrid was still there, especially the muggle-borns and half-bloods?

They didn't know, remember? Again, from your own words:

But the headmaster, Professor Dippet, ashamed that such a thing had happened at Hogwarts, forbade me to tell the truth. A story was given out that the girl had died in a freak accident.

There were rumours, certainly, but no more than that. Only the school staff, school board and those at the Ministry knew for sure.

(Both quotations from Chamber of Secrets, chapter 13.)