I read somewhere that if you want a few great photos, take primes. If you want many good photos, take zooms. I think your plan is solid. Using the 24 to 120 will most likely be close up. Those photos do not typically require the highest autofocus capability. Have a great time!
The 400/2.8, 600/4 are much brighter and faster focussing than any zoom. However, they are also vastly more expensive and larger/heavier.
I am lucky enough to have been on safari in a vehicle (always based on the landcruiser conversion) with a driver and spotter/student for the last 3 trips (most 2-3 weeks long) and so handling / carrying the gear and ensuring it is safe has been easy. The same cannot be said for those who shoot from open vehicles or the awful white buses.
You need to take account of the vehicle you will be going in on game drives, how full it is going to be and quite literally how much space you will have to swing a lens.
Lions are most active around sun rise and sunset -- if there is a kill then you can have some interaction. Sometimes lions are still chasing, say a buffalo, and you can come across action - but during daylight (except when there are crossings in the mara) most often the only action is opportunistic (some prey "volunteers" to be lunch), mating, or some battle that has been going on over night.
Leopards also more often hunt in dusk - but they tend to come out of their dens a few hours before they hunt to sun/warm themselves atop of termite mounds. Large males patrol their territory and can be seen during daylight.
Cheetahs are almost exclusively day time hunters - sure early morning and late evening but almost anytime - as far away from other cats and hyenas as they can - both will steal their kills and attack them. When we shoot cheetah chases most often we are standing back some distance to give the cheetah space to hunt and to not disturb/alert the prey. Once the 6-8 second long chase starts at 70 mph it can go almost anywhere - and typically this is when what often are a large number of safari vehicles vie for the best spots - me I ask my driver to give me a stable shooting platform until we need to move typically to see beyond other vehicles.
So you will need long glass. If you want to shoot cats, small/medium birds and just about all forms of action. Particularly in the Kruger National Park (a public access park), where you have to stay on the roads and any hint of a sighting creates crowds of vehicles.
Other cats - Servals etc.. are much smaller and spotting them can be very challenging.

Primes and the zoom for closer/wider I put on my 3rd body (used to take this photo) - not the panning plate I use on a beanbag. The zoom is the Z100-400. I also carried the 24-120/4, which I used for this shot and some "landscapes"
Just a couple of examples and no ultimate action shots. Mainly close in shots.

Young female leopard fleeing from a pack of hyena who had surrounded the tree where she stored her dinner. NIKKOR Z 400mm f-2.8 TC VR S -¹⁄₁₂₅₀ sec at ƒ - 5.0 - ISO 450 - ²⁄₃ EV
[ATTACH alt="One of the great advantages of the 600/4 TC is you can "Zoom" in with the flick of a switch. But don't over use it."]4945280._xfImport[/ATTACH]
One of the great advantages of the 600/4 TC is you can "Zoom" in with the flick of a switch. But don't over use it.

Shot with 400TC and Z9
[ATTACH alt="Shot with 400mm on Z9 - Mother moving her cubs as she works to keep up with the pride - each of the 3 cubs was carried about 400m, put in a safe spot and then she went back for the next -- a risky but essential task. She has to look after the cubs and eat kills made by the pride, once the cubs are old enough to "survive" in the pride."]4945282._xfImport[/ATTACH]
Shot with 400mm on Z9 - Mother moving her cubs as she works to keep up with the pride - each of the 3 cubs was carried about 400m, put in a safe spot and then she went back for the next -- a risky but essential task. She has to look after the cubs and eat kills made by the pride, once the cubs are old enough to "survive" in the pride.

Shoot with 400mm onZ9

Leopards return to the kills they have stored in trees to feed -- NIKKOR Z 400mm f-2.8 TC VR S -¹⁄₁₆₀₀ sec at ƒ - 4.0 - ISO 1800 - ²⁄₃ EV
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areallygrumpyoldsod
Nikon and Hasselblad shooter -- wildlife and and --
https://www.andymillerphoto.co.uk/
https://www.flickr.com/photos/ajm057/
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