I think you're being a bit uncharitable here. You've just moved the infinitude/"mysterious magicalness" from talking about real numbers to talking about sequences of rational numbers

That was deliberate. (How was it uncharitable?)

I don't think it's really extraordinary to claim that an undefinable or uncomputable sequence is a bit mysterious and possibly somehow unreal.

It may not be extraordinary, but it's still a confusion. A confusion that was resolved a century ago, when set theory was axiomatized, and the formalist view emerged. The Cantor/Kronecker debate is over: Cantor was right, Kronecker was wrong.

The source of this confusion seems to be a belief that correspondences between mathematical structures and the physical world are properties of the mathematical structures in question, rather than properties of the physical world. This is a kind of map/territory confusion.