Bootleg Grounds and a Light Switch
Several ways to fix it.
Install a new cable run from switch to light.
This is pretty much what you propose.
Replace the existing cable chain with /3 cable w/grnd.
Now, you assign colors as follows:

White = neutral
Black = always-hot
Red = switched-hot

Replace the last receptacle with a smart switch &quot;master&quot;.
It then controls the light.  This switch is in an ungainly location, so you also install a smart switch &quot;remote&quot; in the desired switch location.  The remote talks to the master via either wireless, or powerline signaling.  I'm thinking of the Insteon family of products here.
Now the wire assignments are black=always-hot and white=neutral.
Ditto, but keep the last recep also.
This would be done 2 ways.  One is by blowing out the receptacle box to a 2-gang box, so both recep and smart switch can lay alongside.
The other is by leaving it a receptacle, and using a smart-switch &quot;faceless module&quot; up in the lamp ceiling rose.  (or for that matter wiring the smart-switch &quot;master&quot; to be right next to the lamp) by adding an appropriate box there).