As I said, I have not had a broody raise chicks this time of the year. When I do in the brooder they have a warm spot to go to.
What I do in the brooder is to put rocks in a black rubber bowl so they can walk on the rock but don’t worry if they get their feet or even the down on their bottoms wet. When I have a broody hen with chicks, I add rocks to all my waterers, usually I have four scattered around. It doesn’t bother me or them if their feet get wet, I just don’t want them to get in water so deep they can’t get out. Around here, rocks are free. I’d have to buy marbles. In the winter my waterers are black rubber bowls. In summer the ones in the sun are large white dog bowls so they don’t heat up as much as the black rubber.
What you’re talking about should work. There are two other issues I’d be concerned about. How do you keep the water from freezing? I’m sure you will figure that out.
The other issue is that they scratch a lot, the chicks as well as the hen. They’ll scratch bedding in the water and get it filthy pretty fast. The water needs to be low enough that the chicks can get to it. I used to use plywood but came into some old carpet. I spread that carpet on top of the shavings and put the water bowl in the middle of that. I regularly have to shake the shavings off that carpet but it makes a lot of difference in how dirty the water gets. Since my water bowls are a bit high for the chicks for the first few days and to keep the older chickens from standing on it and turning it over, I make a frame larger than the bowl and top that with plywood. I then cut a hole in that plywood the bowl fits in. That keeps the bowl from turning over and with the plywood a bit lower than the bowl, the chicks can easily reach the water.