Interesting idea. I suppose I dont know why it wouldn't work. All of the bacteria needed would form on the rock just as it forms in our filters. Technically I suppose if you have an undergravel filter the gravel would be 'live rock' of sorts. Bacteria to support the nitrogen cycle colonizes on all porus surfaces in the tank and concentrates on places where the flow rate is highest.
There are definitely issues with nitrates in FW...but many of us don't really see them because we do the same thing that the salties do, we do water changes. Salties have learned that doing water changes or not having filtration like a canister filter will decrease the nitrate production or take as much as possible out...they've further figured out how to decrease nitrates using macroalgae, just like FW tanks can do with plants (many people with planted tanks actually dose nitrates instead of focusing on taking them out with water changes).
My main concern with only having a powerhead moving water past live rock for filtration would be the lack of surface agitation, (leading to lower oxygen levels) and possibly the mechanical filtration...since even SW tanks have skimmers, a FW tank would need something else to take out solid wastes.