Um, I must respectfully disagree. Too hot is not better than too cold. Neither extreme is good. If chicks, regardless of species, get chilled, they'll huddle together and share body heat until the temperature issue is corrected. If they are too warm, they have no way to cool down. Dehydration from overheating sets in fast. The fact that in your experience the chicks stay toward the cooler end of the brooder should be telling you something. Having the "cool" side of brooder at 90-100 isn't hardening them off or exposing them to cooler temps, and having it at 110+ on the heated side is, if I may be so frank, asking for trouble. It may not have happened yet, but it will. I DO agree with having temperature variations from one side to the other. A mother chicken or mother quail doesn't heat the entire space her babies are in either - she's just there and when they feel the need, they duck under her for security or a quick warmup. Then they pop right back out again, regardless of what the ambient temperatures are, and go about the busy and fun work of learning to be adults.
There's a reason that I don't like rubbermaid containers or plastic totes for chicks. There is no air circulation. Just a hot lamp, hot plastic, hot food and hot water. No air is getting in at the sides, and no humid, hot air is being replaced. When some totes get too warm they can off-gas. But I know lots of people use them and love them. I just think they are too much like an Easy Bake Oven, as one of my friends says, and offer little respite and absolutely nothing else to look at but the walls.
Quail can be, and have been, successfully raised using the Mama Heating Method, but since the OP has already ordered a brooder heating plate, that's almost as good. I prefer the heating pad because it's a steadier, gentler heat, but as far as I'm concerned any alternative to hot, bright lights 24/7 is a better option.
And now that I've made everyone mad, I'll scoot out of here.