I thought of that as well. With this approach you hit another structural problem: activities such as solving problems etc are considered appropriate only for "recitations" (i.e., done by TA's) and not for "lectures" (the part where the professor teaches).

Kind of stupid, I know. But it is really impossible to violate the mandate that my teaching should not be "wasted" on doing assignments in class etc.

And yes, when I tried it, I loved it. The teaching time becomes so much more interesting and stimulating. But I can only practice this a couple of times over the semester, as otherwise I will be seen as "wasting" my time in class.