Jason:
I forgot to mention, please, please don't think:
But if there are so many tourist vehicles driving so close to them, then they must be very habituated to the vehicles and it will just be the "open zoo" that Cody described. I might as well stay home and go to the zoo.
There is no way that any zoo in any country can give you the same feelings you will get if you are there in person. The sights, the sounds (lion roaring at night), the smells are incredible. The crowds we all have mentioned are usually as Cody and Pippa stated around the big cats, because that is what tourists particularly want to see. Each place differs. For instance, in Amboseli the elephants are indeed habituated to vehicles and aren't bothered as long as they don't get too close. In Tsavo, the elephants are much more nervous because years ago they were shot from vehicles.
It also depends on how flexible your guide and safari company are. In one place, instead of going on a game drive at the usual 6:15 am., I go at 9:30 a.m. Very few other vehicles to contend with then.
It's nice having the park and its animals almost to yourself.
When we, who go to Kenya often, crab about too many vehicles in one place or another, it is because we have been there before and perhaps seen things without crowds, for instance in off season. Then we compare with the busy season and get frustrated.
I would NOT consider staying at home and going to a zoo. Doesn't even remotely compare to the experience you would have in Africa.