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.