To me, pushups, squats, and cardio don't require stretching, so I personally would not disagree with the instructor on the lack of stretching before those. I am very wary, however, of high kicks at the beginning of the class if I haven't tested my flexibility that day (by doing a few moderate rising kicks for instance or side straddle).
Some think that you really need to get warmed up before stretching. Personally, I think that the data is mixed.
In my TKD class though, there is no stretching at the beginning of class . . . only at the end.