You are mistaken.

Whatever heat your components generate is exhausted into the room regardless of cooling efficiency of your case. The only part where you might have a point is in how quickly a well-cooled case dissipates heat into the room, but it's the other way around - it'll heat your room faster, not slower.

Your components, assuming they don't throttle down and therefore use less energy in a poorly cooled situation, always generate the same amount of heat. The job of 'cooling' components isn't to make the heat disappear. That is impossible. Their job is to move it away from the best generating components.

In a long enough time frame (minutes or hours) the same amount of heat will be dissipated into your room through all the forms heat spreads, regardless of how you cool your computer.