Hagrid is one of the best friends of Harry Potter. We learned during their first meeting that Hagrid is not allowed to perform magic, although he got a leeway to get a letter from Hogwarts to Harry.
Considering Hagrid’s father was a wizard, and Hagrid attended Hogwarts, why was he not permitted to use magic? Because Hagrid was expelled, his wand snapped, and he never got to finish his education.
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Why Did Hagrid Get Expelled?

In the second book, Harry Potter and the Chamber of Secrets, we learned that Tom Riddle accused Hagrid of letting the Slytherin monster out in the castle, attacking students, and killing an innocent girl. And that was the reason Hagrid got expelled from Hogwarts.
Later on, we saw that it was all a misunderstanding. The monster from the Chamber of Secrets was a basilisk, and Hagrid’s monster was an Acromantula, a gigantic spider, which, in fact, was afraid of the basilisk.
Unfortunate for Hagrid, who was always fond of dangerous creatures, he was framed for killing Moaning Myrtle because it was very convenient for Tom Riddle to blame young Hagrid to cover Riddle’s identity as the Heir of Slytherin.
Riddle wanted to stay at Hogwarts during the summer (probably to search through the castle uninterrupted), but the Headmaster said it wouldn’t be possible in the light of the attacks and a dead student.
Riddle could not admit that he knew anything, and nobody would believe that the attacks stopped without cause. He had to find a scapegoat, and it so happened that he discovered Hagrid and his “friend.”
When Did Hagrid Get Expelled?
Hagrid got expelled at the end of his third year at Hogwarts.
Even though Hagrid was expelled for presumably killing another student, we believe that a 13-year-old wizard can’t be sentenced to Azkaban.
However, because Dumbledore was probably the only one who believed Hagrid to be innocent, and because Hagrid didn’t have anywhere else to go, he then pleaded with Headmaster Armand Dippet to let Hagrid stay at Hogwarts as gamekeeper’s assistant.
Why Hagrid Didn’t Get His Wand Back
Once expelled from school, one’s wand gets snapped, and one cannot use magic anymore because of the lack of education and possible danger one may pose to everyone around and oneself, and that was the case with Hagrid.
He got expelled for the supposedly committed crime, and his wand was snapped and was irreparable. Hagrid also never finished school, and therefore he couldn’t get another wand.
Although Hagrid got pardoned after the events in Harry Potter and the Chamber of Secrets, he was already 65 years old, and we can assume that he didn’t want to attend school with other kids, especially that he would have to start at his third year again.
Hagrid was also well versed in life without a working wand. He lived it for 50 years, which is more than his life with a wand (almost three years).
We assume that he didn’t want to get another wand and finish his education because of Hagrid’s age and lifestyle. Plus, he kept the pieces of his old wand, and we have proof he sporadically used them (hidden in a pink umbrella.)
And after his acquittal, he got promoted from gamekeeper to Professor, a Care of Magical Creatures teacher at Hogwarts, and he didn’t need a wand to perform his duties.
In our opinion, Hagrid figured he didn’t need to go back to school and have a wand to be useful and lead his life as he wanted.
Was Hagrid Compensated For His Wrongful Expulsion?
It is not mentioned anywhere in the books if Hagrid got an official apology for being expelled and no mention of apology for the short unnecessary visit to Azkaban.
Still, we have to believe that Ministry and School Board, like it would have been in the Muggle world, apologized to Hagrid for all the false accusations and the hardship he had to endure.
Although knowing the Ministry of Magic was mostly prejudiced against non- and half-humans, they didn’t want to issue anything official. They probably only sent a little letter to Hagrid out of the record.
Unless you count Dumbledore naming Hagrid new Care for Magical Creatures Professor, we don’t think he got any other compensation for the expulsion.
Conclusion
Hagrid had a rather difficult childhood. Starting with his mother leaving him and his father when Hagrid was only a 3-year-old boy. Then his father died when Hagrid was only 12, making him an orphan (just like Harry!) And to top it up, Hagrid was expelled from school for something he never did.
It was bad luck on his part, but we know that Hagrid has a big heart on the right side. Not many wizards competed against Voldemort, even when he was young, and Hagrid had the misfortune of loving creatures that society does not accept.