You can't record realtime modulation into session view clips in Ableton Live, especially if they're native Live devices. You can record MIDI CC with third party plugins if they have their own MIDI learn but that doesn't work properly either.
Maschine is something you might want to look at. It has its own version of session view and can do what you want easily. Upcoming Bitwig Studio has this feature as well.
As I said before, I don't want to record any modulation.
I just want to hit play, and send midi CCs to the virtual synth.
Seriously, this is one of the simplest features in any DAW. Try Cubase, Logic, Digital Performer, Sonar, Reason, Studio One.... try this in ANY other DAW and it just works without doing anything special:
1) write some midi notes
2) hit play and loop
3) move your modulation wheel
Voila, you virtual synth is receiving CC 1.
You don't need any special gear, any midi controller in the world will work.