If faced with the knowledge that physical continuity doesn't exist in the real world at the level of fundamental physics, one might conclude that it's crazy to continue to value it, and this is what Eliezer's post argued. But if we apply this reasoning in a non-selective fashion, wouldn't we also conclude that we should stop valuing things like "pain" and "happiness" which also do not seem to exist at the level of fundamental physics?

Whoa, hold on there. Eliezer's post argued that what what we do care about doesn't correspond to physical continuity. It doesn't say anything about any other correspondences for other abstractions like pain and happiness.