The first part of the question is more to my point (and avoids the precipitous use of such a heavy word as "necessarily"). Here is another way to frame it, making it sound more like a flip side to Question 1, and cordoning off any consideration of necessity in a parenthetical:

Question 2': Why didn't (couldn't ?) mathematics naturally develop in such a way that it resides elsewhere in the lattice of interpretability?"

An answer to the modal version of this question would be inextricably tied up with one's views on the nature of mathematics, to the point of needing to discuss metaphysics. But the non-modal version could potentially be addressed at the level of methodology, or of considerations internal to mathematics. That is the kind of thing I had in mind when I mentioned imagining technical work being able to shed light on the matters.

What we classify as "ordinary mathematics" has a lot to do with our psychological makeup. If we were different critters with different needs, then perhaps our "ordinary mathematics" would live on another chain.

It's a contingent fact from this angle; if we found different things natural or useful, then what we labeled ordinary would live elsewhere. Conditioning on what we do find natural and useful, however, it becomes a necessary fact (in the sense that these preferences determine it). If our preferences are in some way rational (if any rational being would have them), then perhaps we have a "necessary fact" worth explaining.

Just my 2 cents.