Seems like, at some point, it would be lifting a lot of dead weight
It is. That's the whole logic around staging. Of the stack of rocket, fuel, and payload on the launch pad, over 90% of that goes into the rocket, and the great majority of that is fuel -- the payload is tiny by comparison. The more structure you have to drag along, the more fuel you have to launch with. So you get rid of all the excess weight as soon as you can -- i.e., as soon as the first-stage tanks are empty.
The key to an SSTO mission is to create a system that is very light -- the structure, tanks, and engine don't impose too great a mass penalty. You're still going to have to launch with a butt-load of fuel, though.