I love this idea, but in wilderness areas where the key words in the law are "mechanized", you might still get some resistance. At least I recall that was the word used, I will have to dig around in my archives.
I love the paddle wheel idea, and also the gravity dredge. But I have another alternative if anyone is interested. A company called Pursuit Dynamics purchased a steam powered water jet for marine use from an Australian inventor in the 1990's to develop commercially. Instead they developed it into a pump for use in food processing and other commercial uses. This pump is a flow through design, with low pressure steam being injected into the flow chamber from an annular injection nozzle around the outside of the chamber. When the steam hits the water, it transfers momentum to the water in the chamber, but it also contracts and condenses. Due to the shape of the chamber, and placement of the annular steam nozzle, this steam contraction serves to pull water through the pump. It has zero moving parts, and you can place your hand in the water flow at the end of the pump (it isn't hot). It also allows anything small enough to pass through the pump to do so without harming the pump or the object. So this pump could be placed wherever you want in your system.
Picture a small aluminum tank over a fire with a steam line running to your dredge, and a water feed line coming from the creek/river. Or a solar setup with a Fresnel lens or parabolic reflector to heat the water to steam. The steam only needs to be at 60 psi, so no heavy duty tanks or lines are required.
This company recently went under and started buying up online gaming casinos. They sold their marine drive jet to another company, but I doubt they are going to do anything with it. Of course, if CA law now says "vacuum" is forbidden in the device, this system will also be illegal.