I understand it differently and so I’d disagree. As I understand it, the assumption of continuity comes from our mathematics. The Real numbers themselves make the assumption of continuity using infinity which gives us infinite limits. Axiom 7 of ZFC lays this out.

Since physics stands atop the Real numbers and Set Theory, we can’t help but use it in analysis. We don’t have another better tool. In pure mathematics, mathematicians are not so concerned with infinity because it is math for the sake of math. I agree, no real problems there. However, when applied to physics the assumption of infinity evolves into assuming infinity exists in nature. This is a problem because this assumption is not a falsifiable hypothesis. We are breaking our own rules here.

I’d say that we don’t “observe” continuous behavior but rather we expect it because of how we measure and the assumptions that go with it. We don’t have a good reason to think that the universe is truly continuous or infinitely divisible.