I agree, a major decision is whether one goes to a private reserve (which often costs a lot more) or public parks. The Masai Mara and the Serengeti (as well as the Ngorongoro Crater and Kruger) are public. But for the record, there were only about 4 jeeps following this cheetah and, according to our guide, it was a male searching for a female. It wasn't fleeing the jeeps. In fact, my experience has been that the animals to a large extent ignore the jeeps but would agree, it is much more pleasant to be in an environment with only a few jeeps near each animal (like Sabi Sands or in Botswana).
You can hear the cheetah calling for a female in this video:
https://www.flickr.com/photos/60519499@N00/53317749531/in/album-72177720312511178/

If one wants to go to the Giraffe Manor in Nairobi, Kenya, it requires an additional trip back and forth to the airport. Kenya and Tanzania don't make it easy to cross the border from country to country. If one wants to optimize time out on safari (versus spending time in airports), it is best to stick to one of these countries, Tanzania or Kenya. There are differences between the Masai Mara and the Serengeti, but both are amazing and one can easily see the big 5 in either.
PS. We were following a cheetah and when it crossed the border. All the jeeps had to turn around and give up the chase since any driver caught going across the border would lose their license. The only way to cross is to fly back to the capitols and go through customs, then go all the way back to the location.

Tanzania can meet all of your requirements or at least the majority of them. Check out Africa Nature Photography. They led the safari I took in 2019 and it was terrific. Three days in the Ngorongoro Crater and then 8 or o days in different areas of the Serengeti. We saw all of the big 5 plus many other animals and many bird species.

Photo gear should include two bodies, one with a telephoto and one with a wider field of view. Our guides provided sand bags, so tripods and monopods were no needed.

I suspect just about anywhere in Africa would be amazing. Also check he website for Pangolin Photo as they have a wide variety of options.

If you haven't already, find a good travel agent, one who really knows Africa. There are a huge number of places that one can go to with a huge range in prices. So choosing the type of trip will require many, many decisions. The first will be how long you plan to go for. We used Eyes on Africa for our first trip to Africa (South Africa and Botswana). My interests don't seem to match yours, so will through out a few thoughts. I have had "close encounters" with giraffes and elephants in behind the scene tours at various zoos and in places like India and Thailand so the Giraffe Manor in Nairobi (or an elephant encounter) didn't appeal to me though I know some people who went and enjoyed it. Also, when arriving in a city in Africa, my goal has been always to catch the first small flight out of the city and get to a camp as soon as possible. So for example, not staying in the city (ie. Nairobi) at the giraffe manor. I much prefer the spontaneous interactions one can have, for example, in Kenya an elephant visited my tent several times. I have contemplated trying to see the Great Migration but my small experience with that was seeing a herd of wildebeest start running towards the river and then a group of vehicles headed towards the crossing and spooked the wildebeest. Our guide (and we concurred), decided to enjoy the wildebeest in their herds migrating around and not try to see the river crossings. These are the kinds of decisions you can help make if you are in a car with the driver and tracker. For example, following a cheetah on a hunt, etc.
Every trip to Africa is different and depending on your goals (ie. getting photos, personal interactions with animals, etc), many different places to go. Since this is your first trip to Africa, I am sure it will be amazing and unique and you will have a great time! Seeing lions, elephants, leopards, cheetah, rhinos, buffaloes, giraffes, hyena, impala, hippos, jackals, warthogs, etc roaming about is an unbelievable experience.
As far as countries, I have enjoyed all I have been to (SA, Tanzania, Kenya, Rwanda, Madagascar, Botswana) and each are unique. South Africa has lots to offer. So to give one specific area, the private camps around Kruger (ie, Sabi Sands) have a great number of different animals in a small area but not the huge herds one will see in the Serengeti or Masai Mara. And the Okavango Delta in Botswana is a much less crowded area than most (but more expensive). I haven't been to Zimbabwe but other friends I have said there are lots of great camps there.
Are you also looking at photographic safaris? If so, choosing a particular photographer's trip will make some of these decisions for you.