AFAIK, inverters are already fully automatic. It goes full blast while the room is still warm and gradually goes on low power (but keeps the compressor running) once the desired temp is reached. That is what makes it so energy efficient. Therefore, there's no need for putting it on fan then switching manually to cooling mode.

If the room gets warm when it's filled with people, then that means the horsepower rating is not enough for the given space. When considering the specs of the AC, it's not just the room size that's considered. Things that produce heat within the room such as appliances, stoves/ovens (if any), and, yes, human bodies should have been factored in.

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