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?

The relevant distinction isn't between whether something "exists at the level of fundamental physics" or not. The distinction is between whether something reduces to fundamental physics or not.

Perhaps some talk about macroscopic physical continuity can be reduced to fundamental physics. (What you say about widespread agreement on typical cases points to this.) But some talk about physical continuity makes no sense at all within the theoretical framework of quantum mechanics. It's not just that the physical continuity isn't fundamental. It's not just that it doesn't "have a simple definition in terms of fundamental physics". Rather, it can't be given any sense at all. That, at any rate, is the claim in posts like Can You Prove Two Particles Are Identical?. It makes more sense to say that a concept is morally irrelevant if it can't be referred to anything in reality at all.

Pain and happiness might reduce to fundamental physics, while physical continuity (in some cases) doesn't reduce to anything. There would then be no inconsistency in selectively discarding physical continuity but not pain and happiness.