The motion along the y-axis is absolutely not irrelevant. You are right to feel bad about that answer!
Assuming the balls stay attached to the ground underneath them, the second ball has to do two things differently from the first:
- Travel the distance up and down while changing speed, and
- Cross the valley gap faster than otherwise.
The ball with the complex motion will only beat the simple one if the gains crossing the gap at speed overpower the losses traveling the extra distance up and down. Think about the limit as the width of the gap goes to zero - if the complex-motion ball just stops in the middle to travel up and down there's no way it could beat the first.
I would suggest treatingLet's treat the problem as if the pit's sides were vertical and the ball was on a track that attached it to the edges. You will findThe part of the motion we care about is only the part that involves the answer depends onpit, as the rest is the same. With initial rolling velocity V, pit depth H and pit width ofW, we have that:
- The time spent moving down is $\sqrt{V^2/g^2+2H/g}-V/g$.
- The time spent moving back up should be the same.
- The time spent moving across is $W/\sqrt{V^2+gH}$
Adding them all together gives us: $$ 2\left(\sqrt{V^2/g^2+2H/g}-V/g\right) + W/\sqrt{V^2+gH} $$
Which, as we would expect, reduces to $$ W/V $$
when H is 0.
Clearly, (1) being greater or less than (2) is a complicated situation!
The answer that you remember would be true if the "pit" was not an actual pit, but was instead a potential well whose presence didn't add anything to the distance. If both balls were on a straight track but the second one had repelling magnets on either side, the magnet-ball would win by the argument your question posed.