This looks like a simple mistake of poorly chosen words. In the sentence right before the infinite rise time comment, he notes "Intuitively, the steeper the signal’s rise and fall time, the less influence random noise has in the time domain."
Had he said "the steeper the signals' rising and falling edges" instead, the statement would have been correct and much clearer. I understood what he meant, even if he didn't quite say it correctly.
it may be that to make the signal traverses the decision threshold (vt) instantaneously, in the ideal case you are referring to, both rise and fall times should be zero (not infinite).
Elio P. Balsamo