Descartes begins by doubting everything. This is the beginning of his argument. We can say that it is the first assumption or starting point of his reason, that he can doubt everything.

This is quite explicitly not how he begins his argument. It says there right in the quote you provided that he throws out "everything that can be doubted." He does not say that everything can be doubted. That's why he devotes the entire first meditation to systematically seeing what can be doubted, instead of throwing everything out all at once.

Additionally, Descartes doesn't just doubt anything willy-nilly. He always doubts things on the basis of some reason. This is discussed in the article you linked to:

The procedure of the Meditations is not that universal doubt is supposed to flow simply from adherence to a maxim; to the contrary, the doubt is supposed to flow from careful attention to positive reasons for doubt. Descartes introduces sceptical arguments precisely in acknowledgement that we need such reasons