My brooder is in the coop. I put chicks straight in there from the post office or the incubator, whether the outside temperature is below freezing or very warm. As long as the area the chicks are in is warm enough, has adequate ventilation, and wind protection they should be fine. My brooder is 3 ft x 6 ft and I only heat one part of it. The rest cools down to ambient, sometimes below freezing. Many people would be surprised at how much time they spend in cooler parts of the brooder, thus acclimating themselves to the cold weather.
I think how much room you have makes a big difference during integration. I have an 8 x 12 coop and over 2,000 square feet available outside. Typically at 5 weeks old I just let the chicks out of the brooder and integration is over. They still stay separate from the main flock during the day and at night but they have enough room to do that but no one gets hurt. But yes, I do think housing them in the coop greatly helps integration.