Yes and no.
No: There are lots of very hard open problems which are judged to be of little value by mathematicians and hence garner little attention.
Yes: If a conjecture resists proof for a long time, this can indicate that we still have a substantial gap in our understanding. We project utility into an eventual closure of this gap, not into the statement of the concrete conjecture at hand. The gain in understanding is what we actually work for. It just turns out that chasing specific results, even if they are mostly dead ends on their own, is useful for orientation.
The (by now solved) problem by Fermat (for all integers a ≥ 1, b ≥ 1, c ≥ 1, n ≥ 3, the equation aⁿ + bⁿ = cⁿ does not hold) and the (still open) Collatz conjecture are perhaps good illustrations of this situation.
In pure mathematics, we reason about a world of abstract objects which are considered interesting ab initio. It may be because they arise directly or often from extremely basic operations, they are connected to many other interesting objects, or that they present special and surprising properties. The importance is basically, there is some surprising, interesting, phenomena which occurs in our world which we don't understand and which we seek to understand. Like science but in the non-physical world.
I think if you create a simple to describe system/construction with a property which is extremely difficult to prove. You are creating an object in our world which is basic but have properties which we don't understand (because we can't prove this property). So indeed I believe it would be an interesting thing to study and be of value. I don't see any problems/issues with this. I don't think the only valuable pursuit of humans is to improve the welfare of other humans. I think understanding the world is also valuable.
Mathematics is largely just smart people working on pointless puzzles, and only by coincidence do these puzzles turn out to have practical applications (it cannot be predicted). Or I guess all the obviously practical problems in mathematics have already been solved -- we're now in a world where math is rarely the limiting factor for human progress (like it was, say, pre-calculus; was FFT the last significant unblock from math?).
It's such a waste of the best human minds. Or maybe the best human minds are actually doing something else, maybe we only notice the handful of Terence Taos, not the hundreds of people of equal brilliance who realized pure math is pointless and decided to pursue physics, rocketry, or quantitative finance.