The motor is not to weak for a supercharger. If that were true, you wouldn't be able to run a turbo.
And yes, the supercharger does run off the crank, but it is also generating boost. At idle I think the Jackson Racing superchargers are putting out like 2 psi. This should negate any parasitic effects it would have on the crank.
And a supercharger like an Eaton type roots blower usually only makes about 6 psi at full boost, the temp. of the air is going to higher because of the lack of intercooling but there are ways to help with that.
And you can make more boost by changing the pulley sizes. You can get more boost out of a turbo easier, becuase you can change the size of turbo etc. to get there. It is difficult to do that with a supercharger.
As far as warm up and cool down time goes for a turbo, there is mixed reviews on that. Some people same turbos are designed to run hard and "be put up wet" and other believe in cool down time. It depends on who you ask. The turbo timer people say cool down is important (gee, I wonder why??). Non-turbo timer sellers seem to torn one the "cool down theory."