Berghaus Yeti gaiters, but you need to have the right boots for them to fit to without slipping off.
The design may have changed now, but I remember years ago (1990s) an outdoor shop assistant demonstrating how to fit those. It took him about 10 minutes to get one of them on and I really thought he was about to burst a blood vessel. With a deep red face and panting like hell, he was straining with all his might to get the edge of the gaiter over the end of the boot and into the recess - and that was with the boot on his lap, not on his foot. They should work well once they are on!.
We very rarely wear them, but for day walks years ago the best gaiters we tried were Paramo (too bulky and heavy for backpacking). We now use the lighter TrekMates. Neither would be of much help with water ingress through the footwear.
If you really want nice warm, dry feet, think about spending some serious money on some seriously good wellies.
That's an interesting idea, at least for short walks in the winter months. Like most people I imagine, I automatically think of the cheap black or green rubber jobs, I'll investigate those.