There are two separate questions here:
what is my standpoint on private education
and what should you do with this particular problem
They seem to me so different that they shouldn't really get mixed up.
I think in this particular situation, I would start with a meeting with the headteacher and your son's class teacher. Be as non-confrontational as you can, but ask what their policy is on rough behaviour, bullying, how their playground supervision, what sanctions are used when children play roughly, how other children are taught to react, if there is a friendship stop for children for children who have noone to play with, if there is any arrangement for children who might feel daunted by the rough and tumble outside. (If you see a private school, make sure to ask the same questions of them).
Reasonable things to expect would be:
that all children are taught that they must tell an adult if somebody is hurt or frightened
that there are adults available at all times
that children playing roughly would be withdrawn from the situation immediately
that lessons are supervised so that the children are kept under control
that teachers are extra aware of a child that may be struggling to settle in and that they have thought out some technique for handling the situation
Some sort of buddy system can also be very helpful.
At my dc's infant school two classes were taught in the same (largeish) classroom, so you had between 50 and 60 children in there- very noisy, but the social training was immaculate- the kids really learnt to look out for one another. This is still doing them good when they've moved on to junior school; they know when and how to intervene.
I would have been very loath to withdraw my children from the local council school as it is such a very strong bonding force in our local community, but that may well be different where you live.