Electronic – Why do we not use true sine waves in power inverters instead of PWM

You say you're not familiar with digital oscillators yet, but I think it's worth looking into. Using a microcontroller with PWM output you can input the signal directly in the amplifier's format. So you won't need an A/D conversion at the amplifier's input stage.
The code for the microcontroller is very simple. Basically it's like this: you need a sine lookup table, and a timer interrupt generating your sample clock. This can be 5kHz, for instance, then you get 100 samples per cycle, which is enough for an inverter application; the transformer will work as a low-pass filter. Every 200\$\mu\$s you take the next value from the table and set the PWM register with this value. The output will be pulse-width modulated following the sine pattern.
This means that your class-D amplifier doesn't need the input stage with the triangle wave generator + comparator (this would otherwise be you A/D converter), so it only consists of the driver stage for driving the output MOSFETs.
You get a crystal precision frequency virtually for free.

PS: a center-aligned PWM signal will give you a cleaner sine approximation, but many microcontrollers only can generate edge-aligned PWM. This Freescale controller is available in an SOIC-8 package and is an example of a microcontroller which can do center-aligned PWM.

image from this document

To be efficient, the output amplifier has to be switching (Class D), but other than that you're not missing anything.

It's easier to do this with a microcontroller synthesizing the class-D drive signals directly (which become sine waves after filtering) than with an analog oscillator (which won't run at a crystal-controlled frequency without more parts, and requires AGC to get a stable low-distortion output level), so that's generally how it's done.

I would expect you might find some very old designs (when microcontrollers and microprocessors were relatively expensive) using just that method.