Why 2 pole induction motors don't use auxiliary windings while running (while others do)?
The choice of starting scheme for a single-phase motor is primarily based on the desired torque vs speed curve. More specifically the scheme and capacitor value determine the available starting torque (zero speed, locked rotor), accelerating torque (0 to 80%+ speed) and peak torque. The torque required depends on the load characteristics. The torque vs. speed diagrams shown below illustrate the differences among the schemes.
The number of poles does not have a direct bearing on the scheme used. The number of poles determines the motor's synchronous operating speed and thus the normal operating RPM.

Figures adapted from Fitzgerald, Kingsley, Umans, Electric Machinery 4th ed