The opposing argument there is that the hope is that solving these problems reveals other interesting maths knowledge along the way. Finding a counter example all but ensures that won't ever happen.

No, it opens up a whole new suite of questions. Now we can ask, for example: what conditions do we need for the result to hold? What dimensions does it hold in? And many more...

I guess the question of why counterexamples are so rare and/or hard to find is still left to be investigated.