I've often pondered flying near tornadoes or supercells as I'm also a student pilot. My main concern would be the extreme windshear. In a downdraft an aircraft loses lift and is pushed towards the ground. If you are flying quite low (let's say because the LCL's are under 1000 and you are trying to stay under the base), I think you run the risk of crashing if you fly under the rear flanking downdraft. Updrafts might not pose an immediate risk of a crash, but you also run the risk of losing control or being vaulted into the storm's core.
Core punching is absolutely out of the question, as severe hail doesn't just end a chase like it can for a ground based chaser, it can disasterously end a flight as well. This is less of a problem in a plane though as you aren't bound to the road grid and can simply fly around the precip core.
I've seen videos of aircraft flying under the rain free base without mishap, I think this is a risky manuever, however, as you never know when you might hit dangerous wind shear or turbulence. Supercells pack more than enough power to exceed the stress limits of a small aircraft, and you could always be knocked into an unusual attitude that could be difficult to recover from at such a low altitude.
That being said, I think if you stay ahead of the rain free base, you could effectively and relatively safely chase in an airplane. I think the view would be unsurpassed and you could avoid many of the problems of ground based chasing, such as limited road options, poor road quality, hill and tree obstructions, chaser hoards, and the difficulty of playing catch up on a fast moving cell.
In the videos you mentioned, the pilots are often not directly in harm's way. The news helicopters are often out from underneath the base of the storm, and have cameras with huge telephoto lenses. Waterspouts are also much weaker than a strong supercellular tornado. Their parent towering cumulus can still create turbulence and wind shear but not to the extreme levels of a mature supercell.
http://www.youtube.com/watch?v=mIAO1ZPK4vo