I think there's a few problems with the argument you've formulated and attributed to Descartes.
I'd say one issue that happens here is what "you" means arguably shifts throughout the argument. 1 maintains that "you" engage in acts of imagining without "your body" but it's not clear that the two uses of you are the same here. This unhinging where there's a separate thinking part is key to making the argument work, and it's also a problem insofar it is the assumption of dualism.
A second issue is that it's (now) a contentious claim that conceivability = metaphysical possibility. But even if we grant that, it doesn't at all follow that merely because something is metaphysically possible that it is actually so or genuinely "separable" in any lose way. To give a trivial counterexample, it's metaphysically possible that the same person as me could exist without either his left eye, right leg, part of his heart, part of his kidney, or part of his bone marrow, ad infinitum. It doesn't at all follow that this mean the same thing can exist if all of these things are absent. (This is a variation on the ship of Theseus being used as a reductio).
If the physical example is unwarranted, we can do the same thing with cognitive capacities. Descartes promulgates a list (once or twice in Med 2 at least; again in Med 3 I think) of all the different faculties of mind. Remove one and maybe it's the same res mensa; remove all and it's nothing. But if conceivable removability means not just non-identity or part-whole identity but that the part is not constitutive of the whole, then it turns out that none of these faculties are necessary for the "you."
Of course return to the very top line here. I don't think this is quite what Descartes argued or even the best way to formulate it. I think Descartes' argument is stronger than what you've laid out (though still flawed) in that Descartes depends on two things to motivate his position: (1) the vicious cycle of doubt and (2) the circular relation between the proof of the self and the proof of God. A good source to read on this is John Cottingham.
The vicious cycle is that either cognitive faculties are trustworthy -- in which case we can trust their judgment that a thinking thing exists and engage in these acts or they are not trust-worthy in which a thinking thing is engaging in the act of doubting them and therefore exists. (The thinking thing doubting that the thinking is doubting does not lead to a bad infinite regress because the argument does not hinge on the content of the doubting).
It's also important to note that Descartes' all-things-considered view (not the caricature people present of him) does not deny knowledge from the senses or the validity of the senses, it just puts this beneath the divided line.