From a former student who always has interesting things to say on Internet law topics:
The Sony decision did not create an absolute right to time-shift. The Court recognized time-shifting to be fair use in that case–by a 5-4 decision, not a slam dunk–in part because the recorded programs had originally been broadcast for users to watch them once, and most Betamax users watched their recorded programs once, shifting only the time at which they watched them. (Another reason the Court ruled for Sony is that the plaintiffs represented only a small portion of copyright holders affected by video recording. Other copyright holders–the sports networks, PBS, Mr. Rogers–did not object to their content being recorded by Betamax users.) YouTube is not perfectly analogous to the old broadcast networks, Sony does not fit perfectly.