This may sound stupid but take a look at off road mountain bike competition shoes.
Waterproof, no laces, velcro straps, aggresive sole, removable toe studs, i know they are too rigid but then they meet a lot of the criterias that we need.
I cycle axle deep in mud and have good grip when i need to get off and push thru the worst bits, but at the end of the day my feet are dry and the shoes just get hosed off when i wash my bike down.
just a thought.