Considering your requirements it's always going to be a compromise.
I grew up in South Africa so I'm somewhat knowledgeable on South African National parks.
1) You could book a chalet in one of the national parks there. Because the national parks are run by the government, prices are pretty reasonable. I'm familiar with the Kruger National Park and Pilanesberg National Park.
www.krugerpark.co.za
Pilanesberg is home to the Big 5, along with an abundance of flora and fauna. This Game Reserve in South Africa is every wildlife enthusiasts dream.
www.pilanesbergnationalpark.org
Each of these parks has different camps where one can rent accommodation. Many chalets or rondavels also have kitchenettes in them with stoves, microwaves and refrigerators. Many camps have stores on site where one can buy food if you want to prepare your own in your own kitchen. If you don't want to make your own, most of the larger camps also have quite reasonably priced restaurants with good food.
These national parks are available for those who want to do their own driving, so you can fly into the country, rent a vehicle and drive to the park. Once inside you are free to go on game drives as often as you want, except night driving is not allowed. You are free to come and go as you please all day long, but you are not allowed to leave the road while driving in the parks. If I'm on a tight budget I would seriously consider doing it that way ( I used to do it often while I still lived there). These parks also offer safari drives in official vehicles with a driver and a guide, and they would be in a position to leave the road and go places you cannot with your own vehicle. These excursions can be booked in the park at an extra fee, and there will be at least one or two such guided drives every day.
2) As for the private game parks, they offer a fully catered experience and guided safaris. I spent a week in Thornybush in 2016, and again in 2019. Absolutely amazing experiences. You will find links to those private reserves here
https://www.krugerpark.co.za/. I found the guides in Thornybush to be flexible. Often times they asked us what we would like to see on a particular drive and they were willing to take us to where we were most likely to see the animals we were interested in - within reason. Otherwise they would also make suggestions, or alternatively just pick a route and go to see what's available along the way.
In both national parks and private parks the guides in their individual jeeps are always in contact with one another and as soon as there is a remarkable sighting they announce it on the radio, so other guides can also take their guests to that spot. That helps to improve one's chances of sighting the remarkable animals or situations.
So whether you go all out in a private park where everything is provided, or whether you go to a national park where you have more autonomy, in both cases you are likely to have a great experience.