PaulRivers

p.s. You mentioned the clipless five tens being particularly stiff and rigid. The flats are like that, they're like average regular shoes in being somewhat pliable and comfortable to walk around in. With clipless they make the shoe super stiff and rigid, but with flats they can leave being stiff and rigid to the pedal itself.

With shorter flat pedals they were fine but I did find them the most comfortable with longer slightly concave pedals, I use the Dmr Vault Pedals and it's the most comfortable shoe/pedal combination I've ever used on a bike:
https://www.amazon.com/dp/B00MN2ESSI/

A little closer to the topic of this thread there's a waterproof version as well though it's flat not clipless:
https://www.adidasoutdoor.com/FT16.html?dwvar_FT16_color=Black&cgid=fiveten-men-bike-shoes

When I using clipless I preferred to use shoe covers which work better with clipless shoes with a heel. Below freezing, rain isn't really an issue. Above freezing when it's still cool you can wear long waterproof pants and shoe covers.

But when it's warm plus raining there's no perfect solution. If you wear long waterproof pants you'll overheat. If you wear shorts the water will drip down your leg into the shoe. I've seen some people use spd sandals riding in the rain when they commute every day regardless of conditions. Wool blend socks might also work in keeping your feet warmish even when wet. But I don't think there's a solution that would let you bike without overheating and also keep your feet completely dry when it's raining hard and also hot out.