You seem to be focussed on questions of the form, "Why can't an A be a B?" I ask: Why should it? Different things enrich the world and provide opportunities for future cross-fertilization.
Don't try to think just like someone else. The world needs more unique snowflakes. Sadly, and unlike what others tell us, we don't start out unique. But fortunately we can change that.
"How do math people ensure parity between what they obsess about and reality?"
I think what math people obsess about is always reality. The fact that they are obsessing about it at all is a real phenomenon, and we can think about tracing it back to its most immediate cause. That immediate cause is likely to be a very math-like problem in need of a solution. The not-so-math-like problem which inspired that math-like problem may be an awfully long way from where we started searching, but it's still connected.
Some mathematicians (the pure mathematicians) may pursue math-like curiosities without much regard for where they're going. Still, their own intuition and sense of elegance will constrain their discoveries to forms that can come in handy later on. In this case, mathematics acts as a bridge that starts from today's human intuition and elegance and connects to tomorrow's newfound human problems.
Clearly you're looking for an efficient connection, not just any connection. I suspect the only way to find that is to keep yourself in a state of perpetually looking, even as you continue with what you're doing. Be your own JIT compiler--and yet, don't let a JIT compiler be all that you are.
That's my two cents, anyway. I'm not sure this topic is concrete enough to help by much. ;)
