Let me add a few details that tears apart the justification a little more. But I'll have to introduce some speculation about the motives of reusing only part of the first stage, then I can go about tearing them down:
- Perhaps the OP thinks it will be easier to maneuver without the top tank, or at least less likely to break
- Perhaps only recovering the bottom part will require either less fuel or less thrust, making it more viable
I will argue that the real effect of both of these are actually opposite of the intention.
One analogy is that that balancing the first stage is like balancing a baseball bat on the top of your hand. But I think this ignores several aerodynamic factors. The center of gravity is higher with the top of the tank attached, but the center of pressure also moves in a stabilizing mode. If you drop a pipe with a weighted end, then the natural forces will tend to keep it upright. This doesn't change the need for active control once you fire the engines, but you need that anyway.
Also, regarding the point #2, consider the following:
"Grasshopper touched down with its most accurate thus far on the centermost part of the launch pad," SpaceX said. "At touchdown, the thrust-to-weight ratio of the vehicle was greater than one, proving a key landing algorithm for Falcon 9."
When the first stage lands, its thrust to weight ratio is greater than 1. If that sounds crazy to you, you read it right. As such, it would actually be more convenient to have the stage weigh more when it lands. Saving mass at this point isn't a benefit, it's a detriment. You can't throttle the engine down enough to get a perfect hovering type of landing. If you shed more mass, then you'll have to hit the ground harder, and with less margin of error.
TANGENT: This scheme doesn't make any sense for the first stage, true. But maybe it makes sense for the 2nd, or final, stage? If this rocket reaches orbit, then you have a big problem for making it back in through the atmosphere and propulsive reentry is going to be decimated by propellent budget. As such, I think there is growing acceptance that final stage re-usability can not be anything like first stage.
Speaking extremely speculatively, a fix for this might possibly be to reenter some of the final stage's hardware inside of a capsule and heat shield sent up separately. The idea is that you tear down the rocket in an orbital workshop and prioritize the most valuable and easy to reuse parts. If you can pack those parts from several launches into a single capsule, then it could possibly make economic sense. This would be a form of re-usability for the engines, but not the tanks.
I don't know if it's a very good idea, but it could be coherent.