I understand your point: Setting a threshold of probability for saying that one believes P imposes a distinction that probably shouldn't make a difference in one's actions. Therefore, one plausibly shouldn't impose such thresholds at all. However, I don't understand this line

Belief in an outcome N means that you set p(N) = threshold(p(N)), so now p(N) is 0 or 1.

It would certainly be absurd to set thresholds if doing so had this consequence. But why does it?