There's a third option. The definition of uninteresting we're using may be flawed. Here's a quick stab at a more rigorous approach:

We could start by defining a set of "all numbers that are uninteresting other than by membership or position in this set".

That describes the set the proof naively called "interesting numbers" without the contradiction.

Then we could create a second set with all members of the first set except those that are interesting because of where they are in that set (smallest, whatever). This is a new version of "interesting numbers" that approaches the version in the original proof but is, in human terms, less interesting. As you said, "Being the least member of the set of uninteresting numbers isn't as interesting as we assume."

We could repeat that, making a sequence of sets that approach the definition of interesting in the original proof, but the definition of each set is progressively less interesting in human terms.

Then if we really want to be rigorous, we could talk about "first degree interesting" (what most people mean), "nth degree interesting", or "asymptotically interesting", but the last one is an empty set.