[ygra]

On the first stage every last little bit of weight counts since it only weighs the rocket down. They already lower their payload to orbit by about 30–40 % when they land the first stage. That stage has to fight most of Earth's gravity and atmosphere so accelerating at all takes a lot of effort. So adding weight to the part that is only needed for accelerating the rocket through the atmosphere lowers the total payload considerably. It's bad on the second stage too, but not that much (but even then, they shed the payload fairings or the Dragon nose cone when they're no longer needed).

Adding another (different) engine also complicates things. Currently they only have a single engine type for the whole rocket (plus thrusters), the one on the second stage having a different nozzle. This greatly reduces complexity and thus cost and risk.

[zardo]

You've got the effect of additional mass on the stages backwards. Additional mass on the first stage is bad, but additional mass on the second stage reduces payload in greater proportion.

Mass on the second stage is on the rocket during both 1st and 2nd stage burns, while you get to leave 1st stage mass behind for the 2nd stage burn.

[ygra]

Good point; my bad.