All info provided is great. We have decided to do a 4 day Safari out of Johannesburg to Kruger. Now, we just have to decide on the company (and that is not easy).I'd be inclined to deal direct with a safari lodge.
Booking.com is a good place to start - they give very full information, photos, and genuine independent reviews (only those who booked can review) for each lodge - but there are doubtless specialist agencies.
Also consider flying from Jo'burg to the nearest airport such as Eastgate (aka Hoedspruit) or Kruger/Mpumalanga, which avoids a 4 to 5 hour drive from Jo'burg. Transfers from local airports are usually included in safari prices.
Just from limited personal experiences, Klasserie private reserve gave us very disappointing sightings whereas Sabi Sands private reserve gave my other half and daughter more and better sightings than one could hope for. I think that's down to the topography, or mebbe just a matter of luck
It seems to us that the higher prices are on reserves where there is abundant game, though you are treated royally at all.
Do be sat down when you check prices, they're eye-watering. But they include at least two game drives per day, all food and sometimes drink.
Whether at a private safari lodge or a NP rest-camp you can't go out in the dark, unless in a private lodge on a jeep night-drive. This is in order to avoid affecting the diet of the lions
If you are happy to rent a car, do consider just the Nat Park itself and stay in a bungalow in one (or more) of the rest camps or at independent lodgings outside the NP near one of the gates.. That's massively cheaper than the private reserves, and we had great sightings in just a day and a half.
Downsides are that you're on your own, though just like Yellowstone you can be lazy and just watch for knots of other visitors' cars at the side of the road/track or you can book a shared safari jeep. And you're not allowed to leave the designated roads & tracks and plough off into the undergrowth, either in your own car (you wouldn't want to) or in the safari jeeps.
The accommodation at the rest camps is quirky but pleasant and there are plenty of facilities like restaurants, pools, etc. or you can self-cater. Gates to the NP are closed at (6?) pm & day visitors have to be out by then, gates to the rest camps close at (6.30?) pm and residents have to be back in by then. Gates re-open around 5am.
IMHO 3 days (first afternoon, 2 full days, last morning) is plenty.
If you choose to drive and flight timings allow, you can rent a car at Jo'burg (O.R.Tambo) airport, head to Pilgrims Rest & Blayde River Canyon and an overnite at Graskop, leaving just an hour's drive to the NP gates next morning.
Lots of permutations, lots of different folks' opinions which is best.