The special from Mannheimer Morgen is only about schools from fifth grade on, so it won't help you much now if your son is in third grade.
The only private elementary school about which I have some personal knowledge is the Waldorfschule. A family I knew had two children there from first grade on, one of them liked it very much and stayed to the Abitur, the other one liked it but decided to change into a state school for fifth grade, mostly because of friends going there.
It 's a school in which the parents are supposed to do more than just pay tuition, but also help with fixing things, painting the classroom etc. It's also a school with a special philosophy, either you like it or you don't.
A family member of mine did great in a Waldorf school, but not every family will or can give the support they expect.
It seems to be all in German. If your child does not speak German yet, do get a private tutor. It's mostly up to the teacher, how much help a non-speaking child will get, some teachers go out of their way , sadly mostly won't.
I'm sure there are other private elementary schools in Mannheim, probably run by the church, but I don't have any personal knowledge about them.