We did a private tour and booked directly with a Kenyan operator, driving from Nairobi to Masai Mara. I can't remember who we booked with but can probably find it somewhere. If your budget is healthy then I'd probably go private. Scour TripAdvisor for current recommendations - it's a bit daunting transferring thousands to a random bank account, but there's plenty of reputable local providers. I'd love to go again for longer. If you do choose Kenya, make sure you do a balloon ride.
There tends to be a fair amount of downtime on safari, so do bear that in mind - usually two drives a day. Sometimes the option of a night drive, depending where you go. They aren't permitted everywhere.
Safaris can be brutal, particularly if you go during the migration. Think on this if your child is sensitive to the brutality of nature. I am still haunted by a poor wildebeest who I witnessed fall and break a leg. It then had to lie there waiting to be eaten alive or die first
In South Africa I've been to Kruger, which is a really different vibe to Masai Mara (in my opinion). I also spent time in a place called St Lucia, which has great wetland reserves, safari drives (including night drives), turtles laying/hatching... I really liked it there. South Africa allows for other activities if you think a "true" safari will be too slow paced.
Namibia is also a great option if you want a bit of a mix to your trip. You can self drive it easily too and get to see some fantastic sand dunes.
Tanzania and Botswana are on my bucket list but $$$$.