>>1622116Northwave make a nice shoe, but not sure about how waterproof they are, designed specifically for use with flat pedal mountain biking, so the soles are stiffened and grippy but still allow you to feel through them, unlike a lot of the more specialized shoes they also function fine for walking in, there's a white one in the Tribe MTB range (picrel) amd they are at the lower end price wise of cycling focused footwear.
I used to wear tennis style leather sneakers too but as you pointed out the soles just can't cope with the pedals.