Note that the AC-to-DC part is because the controller card will create new AC of a different frequency, to allow the motor to be driven at a different speed. The intention is to have to motor running at a speed very close to the AC frequency.

If you instead use triacs and either throw away some halv-periods or a percentage of the initial part of each half-period, the motor will no longer be powered by a sine wave. That is not an operating mode the motor is designed for.

Another thing is that the motor is optimized for a specific frequency. The motor doesn't just have a DC resistance. It also has an inductance, and the iron in it has a limited capacity to store magnetic energy. If you reduce the frequency it will run slower. But each half-period will have more time to charge magnetic energy. So you have to be prepared to reduce the amplitude of the signal at the same time as you are reducing the frequency. This is similar to transformers in DC-DC converters. These transformers are tiny compared to a transformer for 50/60Hz, because at a switching frequency of 100kHz, the transformer don't have to be able to store so much magnetic energy. It "pumps" less energy each period, but instead do it more often.