What's wrong with this PWM-controlled constant current sink?
I would not call this a PWM controlled constant current sink. It is a linear current source that is being controlled by an analog voltage. The analog voltage is just being generated by a filtered PWM. It's heating up because it's linear current source, not a switching current source. The 15A rating of T1 isn't for operating as a linear device. It's for operating as a switch. That 15A rating is meaningless here. If you're expecting the efficiency of a switching circuit, don't because it isn't one. It is inefficient (and hot) in the same way a linear regulator is hot.

So it's operating as expected. You've basically built a linear regulator but have sized it according to the efficiency expected of a switching regulator. For something like this to produce 15A, you need parallel transistors with balancing measures, mounted to a heatsink and possibly a fan.