Thrust and gravity work against drag and lift. I would say the elevator is not your speed control because it controls angle of attack. Increasing lift, increases drag, and it is that drag increase that slows you down. If an aircraft designer could work out a way to create lift without drag they would surely do so.
Variation in drag is why the elevator effects speed, but that is not it's primary purpose (which is attitude control). If you want to go faster AND remain straight and level you need an excess of thrust over drag to enable an acceleration (since lift and gravity need to remain balanced). You will get that excess from both increasing thrust and reducing drag, you won't get that soley from your elevator or solely from your engine because a change in one will effect the other. We all know that the four forces I've written about above are not uniquely seperate in the real world.