I think this sentiment is coming from the same place as that of people who oppose more "pure" scientific pursuits in general. You can ask "what good does it do

I think this sentiment is coming from the same place as that of people who oppose more "pure" scientific pursuits in general. You can ask "what good does it do me that some guy was able to go to the moon?" or "who cares about theorizing about black holes, they're really far away", but ultimately pursuing these goals is what got us such crucial everyday stuff like GPS. Sure, you may not care about the goal (even though I think they're worthy goals), but you sure as heck benefit from the journey that got us there. Similarly, chasing mathematical conjectures, even the ones that turn out false, sets us off on journeys on which we discover/invent ever more sophisticated and powerful mathematical tools. And these tools have an amazing tendency to give us insight into broader problems, and help us prove other theorems, even if the conjecture we were originally chasing turns out to be false. It is in this way that chasing conjectures gets us closer to the truth, regardless of their own truthiness.