There is no problem doing cardio and workout on the same day. Most people freaked out thinking that cardio would cause them to lose muscles.
You can cut and build muscles at the same time. Most people couldn't do it because they had no patience. Muscles grow very slowly. As they lost body fats, they thought they were losing lots of weight and afraid that muscles wouldn't grow. So they quickly resort back to their bulking diet and as their body fats grew back, they felt happy because they thought they were gaining muscles.

This happened to most of us including me. This is what I call cheating to look big meaning you need those fats to look big.