In my experience - it's the reverse. Expensive private schools were quick to expel students because as much as they liked the money they liked having good academic results they could boast about much more. It's the basic run of the mill public schools that can't expel anyone because the student has to be in education somewhere and they might be the only school in the catchment area, so there are no good alternatives.

I went to a school decades ago that was both small, and highly effective at explusion. I can't say that this successfully led to improved academic outcomes however.

Of course, none of this addresses why there are behavioural problems in the first place. A shrink alone may not cut it, especially if there is a wider toxic culture in the school which helps create bullying.