Thanks for reading the article and your comment, Isaac! Sorry if I did not elaborate enough on Descartes's part. Here's the elaboration:
Yes, Descartes did spot on some important stuff, as I state in the article that Descartes was extremely close to starting phenomenology, while he missed it. "Cogito, ergo sum" was proposed by Descartes as the starting solution for his skepticism(for reference, he heavily developed it in his Meditations on First Philosophy). "Cogito" is a phenomenon, in a phenomenological sense, even though Descartes did not focus too much on its "immanence," the absolutely given character as a phenomenon. Descartes did not get this wrong from a perspective of phenomenology, but rather he did not care on focusing merely on this and did not start a whole school of philosophy in investigating this. Descartes went on to prove God and the immorality of soul later on, which is what I meant by "missing the chance to start phenomenology," as well as basically all the philosophers from Descartes until Husserl. Phenomenology might seem to suggest the immanent aspect of the self, but phenomenology rather did not care too much about this neither. The major goal for Husserlian phenomenology is to revive, or to simply do, metaphysics through phenomenology, which I can write an article on how to do later. For instance, we can draw the essence of "red" through phenomenology, without concerning whether red actually exist or not, as we start from experiencing the indubitable phenomena of experiencing "red" to compare instances as to what extent can we still call one thing red or other colors. In fact, Husserl thinks that one can draw universal invariant structures of phenomena which diverges extremely from the previous philosophers like Descartes and Kant who despise human experiences as random and unreliable with a lot of illusions and so on. You were right about changing "I experience, therefore I am," as thinking is a part of the mental process which is a type of experience, or phenomenon. It would just be nice if Descartes realized this point, but he didn't. He focused too much on the processing of "doubting", without realizing that it is actually the human experiences, the phenomena, that are actually indubitable, and went on in proving that the edge of doubting is the existence of human beings as in the process of doubting their existence they are already existing. It would be nice only if Descartes realized what you realized-it really would.
Heidegger swings a bigger hammer on Descartes, as he theorizes that Descartes even got the starting point "cogito, ergo sum" wrong. Heidegger would argue that, at least as I understood, the phrase should be rephrased as "sum, ergo cogito." His concept of "thrownness" describes the human condition in which individual existence of Dasein to be "thrown" in the world and suggests the naivety of Descartes's reasoning, which dates all the way back to plato's time. Human, or Dasein if one wants to be really Heideggerian while talking about Heidegger, finds itself in the world: we are born existing. Thus it is naive for Heidegger to reflect ourselves either outside or prior to the world in which Dasein is thrown in. From my understanding of Heidegger, it is illegitimate in asking such questions like whether there are something at all rather than nothing, or whether I exist or not, because we are always already in it, the meaning, the universe, the world: things that are. It is only from within that one can deal with anything. So there is no "therefore I am," because I always am. And only because I am, I think. Instead of standing from outside trying to figure out being like traditional philosophers, Heidegger would say that we can never be in that position, because we are always in it: already inside those questions. The Cartesian tradition is therefore a long tradition of wrong enquiries into being.
I'll try to update more! Thanks for the support!