I don't know enough to suggest that your son has a sensory processing disorder but your description fits my grandson in many ways and he has a sensory processing disorder. He's been in special ed because of speech difficulties. By the time he was 5 he was having behavior difficulties. He was too rough with other children. He wasn't aware of his own strength. He was unable to relate to other kids verbally and his frustration began to show in physical behavior such as pushing, hitting, kicking.
He's been in a therapeutic school since the first of the year and he's noticeably improved. He's 7, nearly 8. He was in a therapeutic preschool when he was 4 and was helped there too. In between he was in special ed and that wasn't enough structure or treatment for him.
I suggest that you have your son evaluated by a developmental pediatrician. He may have a disorder that is making it difficult for him to be aware of his body and how to use it effectively.
At the very least try some counseling with him. A counselor specializing in children may be able to help you learn different skills that will help you work with him. When my grandson was first diagnosed a social worker came to their home once a week and taught my daughter a way of disciplining that worked with both of her children better than the ways she was using.
Why can't you let your 8 yo play on the playground without her preschool brother tagging along? It makes no sense to me to put the 3yo with big kids. He's at a different maturity level, has fewer social skills, and doesn't even like to play the same games. Seems to me that you're setting him up for failure when you know he doesn't get along in groups of children on the playground. Each child needs to be treated based on their maturity level and abilities. Three year olds should not be expected to play with 8 year olds and vice versa. Yes, as siblings they will play together some of the time but it should not be an all the time requirement.
My 10 yo daughter is allowed to play on the apartment playground by herself but her 7 yo special needs brother has to have an adult with him most of the time. Since January and attending the therapeutic school he is able to play on the playground by himself some of the time. Whether or not he needs supervision is based on his behavior and his needs.
I suggest your son has special needs, as you have said, and needs to be better supervised on the playground or possibly not put into the stressful position that playing in groups puts him in. Special needs means special treatment.
Being the mom of a special needs child is difficult. Other moms won't understand, especially if you're not able to hear them and sympathize with the difficulties being around your children cause them. If you make an effort to understand them, and stop being defensive, I suggest that they will be more understanding of your situation.
Your so what happens sounds like you're very discouraged. Please get some help and some support so that you can continue to be an advocate for your son. You learn ways to manage his behavior so that you can teach him how to manage his own behavior. Every parent has to do this when their child is 3. You have an extra challenge. Get professional help!