Can/Should DC-AC Inverter Be Used To Power Inductive Loads (AC Motor)?
... after many hours of searching it seems that inverters are not intended to operate on inductive loads such as multi-phase motors. Why is this the case?


It's not. Fortunately.
While motors present some challenges compared to eg purely resistive loads, they are not especially difficult to drive and it is common to use inverters for this purpose where the use is warranted. The main application area is the variable speed operation of conventional mains operated induction motors - which are usually fixed speed devices when conventionally driven. 

Many AC motor driving inverters are available  - either
from AC mains - to DC bus  - to AC out, or
from low voltage DC - to HV DC  - to AC out (less common).
The main target is 3 phase induction motors as these are industry standard, low cost per power out compared to most alternatives and make good use of existing power supply infrastructure.
Single phase versions exist (I have several) and many (but not all) 3 phase drives can be used for single phase motors. A 1 phase motor still needs a 2 phase drive as both leads need to be able to be driven above and below the sine wave midpoint (or a bipolar supply with high and low side switches is needed - which is essentially the same requirement).
So, unless 1 phase motor is an essential requirement a 1 phase inverter is generally less attractive than providing a 3 phase inverter and motor.
3 phase motors "work properly" as induction motors whereas 1 phase induction motors are a compromise as there is no 'true' rotating magnetic field to follow and this must be provided by the motor design. 

A block diagram of a typical circuit is shown below.
Input at left is here from fixed frequency mains AC, but could be DC or AC from an inverter.  A suitable "high" voltage bus is formed at a voltage somewhat above the peak AC voltage to be supplied. About eg > 330 VDC for 230 VAC out and > 400 VDC for 3 phase 230 VAC.  3 pairs of electronic switches rebuild a PWM waveform which can be filtered to give (here 3 phase) sinewave at variable frequency.

Motor connections shown for 3-phase (star connected) motor and single phase motor. Bus voltage can be lower for 1 phase and for delta-connected 3-phase than for star-connected 3-phase but does not have to be. 



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Here is how PWM can be used to make one AC phase.
The waveform at top of image shows a PWM waveform chopping a high voltage supply. Here bipolar supplies are shown with negative OR positive PWM relative to center ground BUT unipolar rail to rail PWM can be used to provide any voltage from one rail to the other.
The lower waveform shows the resultant poorly filtered sinewave.