There's a major problem with paddling a SUP board down a fast river, and it is that falling overboard is not an option (if you're a sensible guy/gal) but it's very likely to happen, sooner or later. Once you lose balance while paddling standing, you need a 'Plan B', namely a way to regain your stability, or else you go overboard, and SUP boards don't offer such Plan B.
However, W kayaks do offer it, in the form of a 14 inch high, longitudinal saddle (high seat) that you straddle with your legs while you're paddling standing, but you can always fall down on and regain your stability from the Riding position (think riding a bike, or a horse, or an ATV) which offers more control than the traditional kayaking position know as the L position.
Words are not a good means for explaining such things, so here's a link to a blog page with a demo video that includes showing how to "fall", regain balance and control instantly, and get up to a standing position immediately after, without using any accessory: The World’s Most Stable Fishing Kayak In Reality | WAVEWALK FISHING KAYAKS
Having said all that, I'd like to add that SUP paddling in whitewater is too risky to be considered a recreational sport, even in a W kayak.
I would also like to use this opportunity to add that SUP on a board over flat water isn't a viable option either, because a board doesn't offer means to paddle from a lower position, so there's no guarantee you'd be able to go back to where you came from in case the wind picks up, and it's blowing at you from the wrong direction - and your entire body acts a sail...