> If the NYSE went down, wouldn't most trading stop because of people being frightened about using old mismatched prices?

Why would you assume that.

Every exchange is a valid place to trade and the sip ensures you always have the correct NBBO

I assumed that some exchange provides enough volume that without it there would be worries that you aren't getting true price discovery. Isn't that a concern with low trading volume? Not that that would necessarily be true, but there would be enough concern to cause some people not to trade. And others to want to trade specifically with those kind of nervous people wouldn't because the people they expected to profit by were staying put. And then it would cascade.