Maybe this could be connected with Anakin constantly losing his lightsaber. In the movie he rarely reacts to this, but if he did show some signs of anxiety due to being defenseless, it could work well with this narrative. After all Obi-Wan reminds him that this weapon is his life.
If we could somehow draw the focus to Anakin’s shock about losing his arm and lightsaber to Dooku, then we would have a good set up to include this plot point about Anakin’s own mortality in III.
Firstly this could be deliberately be brought up by Palpatine after Anakin killed Dooku. Then Obi-Wan could remind Anakin to jump from the elevator shaft a second time and we could change Grievous offense against Anakin’s age to a threat to kill them, to which Anakin response with an insult.
Finally we could change some of the dialogue during the opera scene, so that it focuses on the power to avert death rather than to save others. On top of that we could add “forever” to Anakin’s line about ruling the galaxy.

In conclusion I think that it definitely is possible to add this theme to Anakin’s fall. Not as the central motivation to turn his back on the Jedi, but rather as a contributing factor like his frustration with the council.