With all respect, autoguru.
If someone stepped up to you and asked "I saw a peculiar carriage this morning. It had no horse or donkey but moved at an amazing speed. What was it?"
Would you take that guy seriously? Would you try to explain about combustion engines, fuels, legislation, driver's license, environment and so on. Probably not.
I think that the reason you get no answer is that the question you ask is similar to the one about the "horseless carriage".
There are many fundamental presentations available. Have a look at the ABB, Allen-Bradley, Siemens web pages. There is also a number of FAQs available (look at the top of this page). One of them is faq237-1063.
Your claim that "gears are 98%-99% efficient" may be true for a one-stage gear, but seems to be on the high side. If you need more reduction or variable reduction, you will easily have efficiencies going down to 80 percent or less.
Weight and space requirements increase and maintenance issues are getting more and more important - making accessability another problem in compact machinery.
PWM VFDs have been with us for more than 25 years. Their use is steadily growing and from a humble start in paper mills and textiles they have gained wide acceptance in process plants, ships, HVAC, water and sewage treatment, food processing, electric vehicles, white goods and just about every application you can think of.
As you know, the market is always right. And the market has accepted PWM VFDs wholeheartedly. There is no known technology ready to take over (except for a variant; the NFO Sinus) and market share is growing faster than any other technology. So, market has decided, and I think that you too can benefit from learning more about VFDs.
Gunnar Englund