Would their summer research be an utter failure? Would they be unimpressive mathematicatians?
How much of math success is being lucky enough to stumble upon a tractable problem?
They had started attempting to prove the conjecture. This result could be considered a failure of that proof. Showing they’re capable of performing the research makes them good mathematicians.
I believe the answer to your last question is “lots.”
These seminars are about deep dives into particular mathematical questions. Maybe including some recent "doable" unsolved problems.
> Stange added that none of this would have happened without the low-stakes summer project. “Serendipity and an attitude of playful exploration both have such a huge role in discovery,” she said.