...and therein lies the problem: he is, by your admittance, claiming an AGENT of thought. Whether that agent is 'something' or 'I' is unimportant.
There is no thinker of thoughts; there is only thinking itself, in the same manner that there is no river that flows, or wave that waves.
There are no separate 'things' in the universe that act; 'things' (and 'some-thing') are only mental constructs. They don't exist in reality.
So if anyone is adding something that simply is not there, it is Descartes, who does not understand that the 'I' which thinks is an illusion to begin with, as it is a self-created principle.
'Cogito ergo sum' is Latin, not French. "Je pense donc je suis" is French.
The correct translation from Descartes:
"I noticed that while I was trying to think everything false, it was necessary that I, who was thinking this, was something. And observing that this truth, "I am thinking, therefore I exist "[cogito ergo sum] was so firm and sure that all the most extravagant suppositions of the sceptics were incapable of shaking it, I decided that I could accept it without scruple as the first principle of philosophy I was seeking."
Wikipedia
And so, my original criticism of this premise was phrased in the question:
"So, when you are NOT thinking, then you, of course, do NOT exist, correct?"
...which reveals one of the basic flaws in its logic.
The problem here is that it is assumed that consciousness is local; that it is contained within an entity called the self, or "I", a separate ego that acts upon the world, and in terms of the topic at hand, a self that lives and dreams and dies.