Here's what you need to understand about chicks and heat.
Primarily and most important, they need two zones of differing temperature. One zone is the heat zone and it needs to be a temperature in accordance with chicks age and feather development: 85-95F the first week, and decreasing approximately five degrees each week.
The second zone is the cool zone, and this is every bit as critical to chicks healthy development and well being as the heat zone. This cool zone should be as large as possible to afford all chicks the opportunity to cool down, thus self regulating their body temperature. It should be a good ten to twenty degrees cooler than the heat zone.
In a room or even a tropical climate with very warm ambient temperatures, the chicks could be in trouble due to no way to create that second zone - the "cool down" areas. You may not need a heat lamp if the ambient temp is 80-90F degrees, but the chicks lack a cool zone. There are chicks raised in these ranges of ambient temps, but they feather out ever so much slower than chicks who have access to cool temps.
Think of it like people warming around a bonfire but there's a huge crowd preventing free movement. The fire puts off a lot of heat, and if you were to remain in near proximity to it because of a huge crowd, there's no way for you to get away to a cooler area to cool off, and the result would be heat stroke. This is what happens to baby chicks when the brooder is too small and it has too many chicks so that no chick can get away from the heat. Chicks go into heat stress and die.
So, a very warm room is going to be a problem, especially if you also have a heat lamp. You may be able to do without the heat lamp, but how will you create that second zone that the chicks need every bit as much?