score:4
As far as I know the only way is via Arusha or by private car.
For the route via Arusha you can book tickets on Busbud (full disclosure: I work on this project). There you can catch a local bus onward, though these run infrequently, maybe not even daily.
I'm not sure what your intentions once reaching the park are, since there aren't likely to be facilities set-up for access without a car. You may be better off booking a safari directly from Arusha.
Upvote:1
An alternative to travelling to Arusha and leaving from there is to get a tour that leaves from Nairobi and includes the Serengeti. This will be more expensive and take longer than busing to Arusha because you'll generally see more than just the Serengeti, but it does match your criteria.
Example 3 (does enter via Arusha)
Example 4 (does enter via Arusha)