This profoundly misunderstands Descartes. Descartes is about proving existence through doubt. If you start doubting everything around you, what is left is your own mind that must exist since it is thinking and doubting. It has nothing to do with being a 'person' as a social construct, or about who I am.
- "I": assumes individuality and that the mind is a local phenomenon.
- "think": assumes cognition and that it's a process of the mind.
- "therefor": assumes causality.
- "am": assumes there is such a thing as existence at our level.
Those are postulates, not demonstrated facts. It's perfectly ok to have those, as long as you declare them as such. However, when you study Descartes, teachers never do so. It's disappointing when the whole point is to teach people to "doubt everything".
There are other philosophies that doubt those postulates. Some assume we are not individuals, but focal points on a collective conscientiousness. Some believe thinking is actually not a creative process, but merely the act of receiving information from another source. Some think we live in a deterministic world, and causality is an illusion our mind uses to make sense of things. Some will say we are not, existence is the symptom of a greater activity like a shadow.
In the end, even something as simple as "cogito ergo sum" can be dissembled to the extreme and debated.
Which is why I enjoy philosophy less and less. It's a never ending source of questions, with very little useful answers for day to day life. It's not practical at all, and my most philosophical friends are also among the most unhappy.
Even antiquity philosophers, despite dispensing practical advises on how to live your life, fall into the trap on not giving any applicable way of reaching their standards.
It's like saying "quit" to a smoker or "eat less" to an obese. True. Practical. Not gonna help.