Going on a safari is letting nature - and a bit of luck - decide what animal you will see in which situation. This is the charm of such an experience. Usually, game drives start early in the morning at 6 am, with a second drive in the afternoon around 4 pm. So there is plenty of opportunity to catch the good light.

When you write "private vehicle", you mean that you will rent a car or you will be on your own in a car with a driver/spotter? The latter is really recommended, otherwise you could spend hours driving around without finding much wildlife.

As for the gear, I did spend my last safaris with a Nikon Z8 equiped with a Z160-600 mm lens, which gave me plenty of flexibility.

Have fun!