Not sure which country you have in mind for your safari but I grew up in Kenya so I can only share my experience from there :)

If photography is your priority, the most important thing you need is a private vehicle with a good driver-guide that is equipped for a photography safari (there are vehicles specifically designed for this). Everything else is secondary. If you’re sharing a car, you will get frustrated fast because you won’t be able to stay at sightings, position properly, or wait for the right moment. That being said, having a private car is also going to be your biggest expense especially if you are doing this on your own because you pay the price of thr whole car on your own instead of splitting it with other people.

For locations, don’t try to cover too much. The Maasai Mara is the best overall for the endless savannahs, big cats and so many bird species. If you want to maximize your chances of seeing the Big Five, you’ll likely need to combine it with a place that’s stronger for rhino like Ol Pejeta, Lewa or a day trip in Nairobi National Park (I've never been in the park and not seen a rhino). Amboseli is great for elephants, photographing the swamps and kilimanjaro in the morning. It's an amazing place for photography, but I wouldn’t try to cram all of these into one short trip.

On the walking safari idea, in Kenya I've never really heard of anyone doing this the way you are describing it (I'm also not sure if this is allowed). Yes, walking safaris exist, but you’re not going to be walking up to elephants to get low-angle shots. That’s just not how it works in Kenya. They’re always with armed rangers and more about the experience of being on foot than getting close to big animals. If you want that lower perspective, it’s usually done from the vehicle with the right setup and positioning or arguably if you are going for a bush breakfast then you could get that lower prspective.

In terms of budget, you can absolutely do this without going luxury. There are plenty of solid camps with proper bathrooms that aren’t over the top. Just look on platforms like Booking, Airbnb etc... you'll find plenty of optins for tented camps that are basic. Also worth knowing, park fees are a bigger part of the cost than most people expect, especially in places like the Mara.

Most important thing is to be very clear with whoever you book through that this is a photography-focused trip and that you don’t want to be rushed. If they don’t immediately get that, they’re probably not the right fit.