Rhett, glad I could offer some extra insight into your planning. Now that you have described what you are thinking of, let me add more food for thought.
Hotels in Northern Tanzania generally do not keep a fleet of jeeps and drivers for day trips like some countries. They are sub contracted out and often have to come from Arusha. In southern Tanzania, your plans can work.
There, most people fly between parks, so game drives are run by the camps. You may be able to self drive. A friend that used to do that area would use a jeep and driver, so not sure how easy to self-drive, and a car may not be suitable.
Even if staying outside some parks, the roads are rough and unsaved. Karatu could be a central town for some of these parks, but again, my point above, bout accessible day trips. The camps in the Serengeti are mostly inside the park, so access is hard. There is more build up around Ndutu, just outside the border of the park, but the road to it is unsaved, and if after spring rains, a real quagmire with black cotton soil (think deep glue) and potholes.
The roads in the more habitat areas include big speed bumps (aka sleeping policemen) which really slow down progress and lord knows how uncomfortable that would be in a car vs jeep. Also, corrupt police. It is not uncommon to be pulled over and ticketed for some infraction...basically for their own pocket money...and, of course, in Swahili.
I appreciate wanting to have some independence. I am similar and like the road less traveled, but found having a car and driver/guide to be the best option. I made the plans, we stopped when I wanted, and took as much time as needed.
You may want to get your hands on Bradt Guide to Tanzania. I used that for my less traveled routes and does speak to more independent travelers. It was published in 2023 so more current than my experiences. Also, their website has updates, and I think trip reports and a forum?
And the Tanzania forum on TripAdvisor are extremely helpful with a vast range of travel types.