In a 3-phase induction motor, the phase windings, placed 120° apart, are fed by voltages phase-shifted by 120° to generate a rotating magnetic field to drive the rotor.
Likewise, in a 2-phase induction motor the phase windings would need to be placed 90° apart and fed by voltages phase-shifted by 90°. In fact 2-phase electric power was generated at the turn of the 20th century at Niagara Falls, obviating the need for capacitors to run motors.
Single-phase motors are actually 2-phase motors, with 2 windings 90° apart, and the phase-shift of 90° being provided by a capacitor.
Your suggestion, to use a high-leg delta system to drive a single phase motor without a capacitor, is not practicable in the absence of voltages having a phase-shift of 90°.
Edited as follows:
There is absolutely no question vis a vis the phase shift of 90° between the line-to-neutral voltages (120 V) and the high leg voltage (208 V).
The only problem would be non-availabilty of the high leg voltage with split-phase consumers.
Availability of the high leg voltage would mean availability of the three-phase system and that would make use of a three-phase motor more valid.