Why do we not use true sine waves in power inverters instead of PWM?
To generate a true sine wave from a DC bus requires a linear amplification system and, this adds up to a lot of power losses (maybe around 35% of the power output). It's as simple as that.
PWM mimics a true sinewave with a fraction of the losses (maybe 5 to 10% of the power output) albeit with some harmonic distortion that is removed using filters (lossless types in the main).

Just compare the class AB amplifier (a linear amplifier with very little distortion) with a class D amplifier (using hard switching MOSFETs and PWM): -

Extracts from this website by MAXIM.