Non-waterfowl when you get them wet yes will not be able to fly. And if they can't touch the bottom they COULD drown. Does that mean they will? Absolutely not. You have a chance of getting hit by meteor, does that mean it will happen? Absolutely not. Thing is on BYC many of the chicken folks are a little different. They sit in the runs with their birds and spend hours talking to the birds, they bring birds inside, they put up heat lamps in their coops in places that may see 1 frosty morning on the coldest day of the year. They take their care to a very high level. Is it bad? Absolutely not. Its just how they feel about their poultry. That's why they suggested covering the kiddie pool.
However I don't think chickens would have any interest in the stock tank. Sure they may walk up the ramp, and may even hang out on the deck, but if they know they can get a drink at the waterer, I bet they'd go there and get one. I doubt they'd lean down and get a drink. I don't think they know they can drink from a stock tank, so they don't think to try it. They know there is water at the waterer so I would believe when they want water they go there. Friends of ours have horses and chickens. They free range the chickens and are often out amongst the pasture where the horses are pecking at road apples looking for bugs. They walk past the stock tank to go back into the run to get a drink and will return back to the pasture to hunt again. Sure they could fly up and get a drink but they totally ignore it.
If it was me Id do the big stock tank and a shelf. I think in the long run you'll be glad you did to save the work your creating by having to dump a pool or other water source you have for your ducks. If you opt to do the kiddie pool, Id get a small submersible pond pump and hook up a hose to it. Put the pump in the kiddie pool, and run the hose out to the garden. That way your not wasting any water. OR you could keep the duck run near the garden so you could bucket off the water and are not going far to dump it. I don't know your property lay out so maybe thats not realistic.