Sure but the point is in a closed system, thermodynamic fluctuations alone are going to take eons to make your house fall down. What will actually get it is that the system is not fully closed, and eventually something from outside the system will give it the activation energy needed to escape the well it's in (e.g. a windstorm causing a tree to fall on it). But it's useful and practical to think of that larger system as decomposing into a product of pretty much independent systems (until it isn't).

The point is wells exist and are common in day-to-day life, including ones that are deep enough that things don't randomly thermodynamically pop out of them. If something is sitting on your desk when you go to bed, you can be pretty certain it'll be there in the morning too. The scale of the well is orders of magnitude larger than the thermal fluctuations, even if eventually a fluctuation could excite your computer monitor to suddenly vaporize. The atoms in your body and all matter will eventually decay to nothing too, but it's not useful to think about that.

And even if it's metastable, as long as the system stays inside the well, the potential is locally quadratic. So you see oscillations around the equilibrium in the meantime.

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.