The first point I'd raise is that any gearing system will have slack (backlash as you put it), although with care I think it should be possible to reduce it to below 0.5 degrees. Even then, though, in a servo system there will be oscillation because of that. My experience (very little, I admit) is that in servo systems pulleys tend to out-perform gears for precision (because there's no slack), but it's a trade-off against power/speed. If I was to build this myself, I'd almost definitely use a pulley or pulleys, because servos handle pulley slippage and elasticity much better than they handle slackness in gears.
My initial thoughts on your need for precision are about momentum. The 60cm diameter wheel surely has quite some mass, and together with the motor spindle, this in my opinion is the dominant factor in your design. So I believe you are right to question whether a simple first order servo will do the trick, because of momentum.
If you can derive (or find on the web) some second order expression describing the acceleration and deceleration required to move as quickly as possible quickly from one position to another, then you are well on your way to solving this. This would involve feedback of the second order, with all the associated headaches. What you have here is the same problem that has had robotics people tearing their hair out for decades.
Failing that, instead of using the motor to brake the wheel, how about a separate braking system to halt and hold the wheel at the correct position? At least the maths would be simplified.
The resolution of your position sensor is not such a major concern. You could consider monitoring the motor shaft instead of the wheel axle, but that might require a zero-position calibration each time the system is powered up. Position sensor accuracy, in my opinion, is an issue to be addressed after momentum.
Lastly, the "why the heck did you" criticism is something I've always tried to avoid, but been slapped with many times. Somebody will always have a "better" solution. Don't worry about it.