Many public schools simply cannot expel problem students.

The school district must educate everyone. When a school expels or excludes a student that student is still a live human that must somehow be educated. They may need to be bussed somewhere. Or to be sent to another school in another district. All of this is very expensive. Never mind that someone needs to administer this.

So in a public school expelling a student increases costs significantly, several fold. In a private school, it decreases costs. The private school just tells you to hit the road.