I didn't figure this out till I watched someone else but you have to place the lever at a specific part of the river to float it to the person at the bottom.

I thought it was flavor text for me stealing their lever and that I needed to use it somewhere else but you literally just give it back to them. I would suggest to the game devs to maybe reword what that npc says or nudge the player in that direction a bit more. Reading the in game manual just made me think floating an item in a river would bring it to the end of the river like I would expect in nature, not that the specific pixel currents suggest that floating items would go different places if you placed them here or there