This is where we need to decide if we're doing physics or philosophy. Physics is about answering specific, fairly practical questions, and the questions tell you what parts of reality you can afford to ignore. If you want to get at the true nature of things in a metaphysically satisfying sense, physics will always disappoint you.

Is it part of the essential nature of a building to collapse? Sure, I suppose, as with all things. But I'm only going to be in this this coffee shop for another hour: to me, for my purposes, it's stable. If instead I were buying the building I'd want a much more detailed model that considered termites and the risk of earthquakes and all sorts of things, but I still wouldn't care about the date of the next ice age that will scour the landscape clean.

One thing to keep in mind is that our mathematical methods might as well be approximate, because we our measurements always will be. Precise calculations on fuzzy data are a waste of time. Not that there's anything wrong with philosophy! But it's not super relevant to understanding where harmonic oscillators are a useful model.