Phillip
A single phase motor cannot intrinsically self start because it is stable stationary when power is applied so not turning torque is to get it moving is produce.
The starter winding, and capacitor if its that sort of motor, are electrically offset relative to the main winding so an unbalanced torque is developed to get the motor going. Once its running fast enough the motor can continue accelerating on its own so the starter winding (and capacitor) are switched out.
The amount of offset torque depends on the winding configuration an capacitor size. If there isn't enough offset torque to overcome the load the motor will be unable to accelerate up to speed so will try and run at the fastest speed it can reach. A failing capacitor has less capacitance than it should so starting and run up torque is also lower.
Higher gear ratios impose more starting loads on the motor so when a capacitor begins to fail the effects on starting will be noticed first in the highest speed gear.
Exactly your symptoms which is why capacitor problems are the common suggestions for things not working.
Clive