I assumed that Alice measures a 1 because the argument would be the same no matter what she measures. Just replace the 1 with -1 and the logic still works. (Basically, "without loss of generality"). Or, I could have some variable X that represents Alice's measurement. We don't know X until Alice actually performs the measurement, but I think the argument should still work if we just replace all instances of 1 with X. I'll have a look at that Veritasium video, thanks for mentioning it.