We already know they have anti-grav / levitation technology; not least because in the second scene of the film, a policeman/bot is portrayed riding an anti-grav motorbike. Is this just an anachronism/oversight by the film's producers?
Are you forgetting the tremendous amount of power that is required to constantly levitate? If you're into the physics, you would need an engine that supplies a dV of 9.81 m/s every second that the wheelchair is levitating. That is a lot of energy.
There is a reasonable argument to be made for anti-grav transportation (on Earth, not for space travel). Levitation means that you do not encounter any friction from the ground (only the air), and therefore incur less speed losses due to friction, thus making for a more economical journey. Additionally: less wear and tear on the vehicle (e.g. the tires and axles), and also less danger from environmental hazards such as wet roads.
But none of those advantages really apply to a wheelchair. It's much too slow, and it hardly travels in a way that the economy of traveling around is an important factor to consider.
Also, an anti-grav motorbike won't be used every second that you're awake, compared to a wheelchair.
A wheelchair is reliable without needed to be refueled.
Also, your anti-grav motorbike argument falls apart even by today's standards: We have electric wheelchairs today, yet not all wheelchairs are electric. The most advanced technology isn't always the most commonly used technology, and it rarely is the only technology that is still used (you're arguing that because anti-grav exists, only anti-grav should still be used, which is a big assumption).
- Wheelchairs do not need to be recharged/refueled.
- We have not seen anti-grav technology being used indoors and in close quarters.
- (Questions) Do anti-grav engines make noise? Can small versions for wheelchairs even be built? Both would be valid reasons to not put such an engine on a wheelchair.
- Even today, electric wheelchairs exist, yet some (I think most) people still opt to have a manual wheelchair. Electric wheelchairs are usually only used by those who are unable to manually operate a wheelchair (if they can afford an electric one, of course)
- Captain Christopher Pike strikes me as an oldschool kind of guy. Though I guess that's not really an answer, since he wouldn't be able to ride a classic wheelchair if no classic wheelchairs should exist anymore (which is the premise of your question).
- There is a thematic argument to be made. Pike's wheelchair shows him as grounded, which thematically connects to why he isn't captain of the Enterprise anymore (grounded in a different sense).