That's not really a little thing. As you said, it's sloppy writing and directing.Right?? Not only does she not use the stick, she actively throws it away. Even if you are confident that the river won't get deeper (why?) of course you'd still keep the stick to look for submerged rocks and just to keep your balance!
I guess on the farm where she grew up they didn't have any rivers or creeks.
Even stupider is that it turns out it doesn't even have any purpose in the plot! It would be dumb if they had her toss it away to make something else work better, but it's even sillier to do it for no reason.
I know it's a little thing. But sometimes little things bug me more than big things. It just screams sloppiness and carelessness.