This is not a mistake. Force = mass * acceleration, so the quicker you lift a heavier mass the more force required. Work = Force * distance so the amount of work you do is directly related to the force used.

You do have a point that slow controlled reps will work out different muscles, but quick reps with a larger weight are also beneficial. With fast heavy reps you will build up fast twitch muscle fibers responsible for explosive strength. With slow controlled reps you will build up slow twitch fibers which are good for endurance and muscle tone.