I did some more research, and it turns out that this is in fact a tried & true concept:
https://en.wikipedia.org/wiki/Inboard_brake
Other answerers were accurate in their assessments, particularly in the difficulty of servicing, which seems to be the main reason why these aren't more popular. Accessing the brakes is more difficult and requires a lift or jack, and replacing a rotor requires disassembly of the axle (a lot harder than removing a wheel).
Aside from that, some pros:
- reduces unsprung weight
- reduces exposure to environmental contaminants
- eliminates torsion forces on suspension components while braking
and cons:
- harder to cool
- increases complexity; non-driven wheels require axles dedicated to braking, and cooling channels/ducts are generally required
The cons seem to outweigh the pros, since the primary advantages are performance-related, but the disadvantages make this approach mostly unfeasible for performance (i.e., motorsports) use. Racing teams tend to work on brakes frequently and may need to change brake components quickly during practice or races, which isn't possible with inboard brakes.
I do wonder if these may make a comeback with electric vehicle (EV) racing due to the decreased demand on friction brakes from the use of regenerative braking. I can imagine a set of pads lasting an entire race, in which case a racing team could swap them between races and eliminate the need for quick access.