Heidegger's fault with Descartes lies in the very first two words: "I think." Heidegger does not believe that the "I" (Dasein/Being) is the thing that thinks (directly at least). In Descartes' world, humans are "res cogitans" ("thinking machines"): the "I" is that which thinks. For Descartes, there is some thing that is doing the thinking. In Heidegger's world, there is not some mysterious being that thinks, but rather the "I" is the experiencing/thinking itself (the "thrown-openness"). This is why Heidegger always uses verbs/gerunds to describe Being.
For Heidegger, the simple statement "I think" cannot possibly be true because there is no "I" behind the thinking. For Heidegger, the analog basic first-claim would instead simply be: "there is thinking" (the "ego"/"I" as we think about it is the "thinking" itself for Heidegger).
The people who think Heidegger's philosophy is compatible with the Dartesian ego are reinterpreting his concepts to fit their own ways of thinking.
It's all so much more confused than most folks, including Descarte, think it is.
He does identify something worth naming, but gets the cause and effect all mixed up. It'd be more sensible to say that I am, therefore I have the possibility of thought, and the possibility of using thought to identify my own being.
Thought being a material process which occurs in-and-of the world.
Would it be? You're already positing that "I am" which there isn't a reason to believe if you were deceived about everything your senses tell you. At best, to start with "I am" simply re-raises the common objection to the cogito which is that Descartes assumes the "I" to begin with. It doesn't get you out of that bind.
Nevertheless, there are contemporary defenses of the cogito, e.g. https://www.jstor.org/stable/41501714?seq=1
That wasn't the inference (at that stage) of the process. He contends that he can imagine himself without a body, but firstly even if his senses deceived him he could still be sure that he exists because he thinks. This isn't such a huge leap, considering, for instance, people with phantom limb syndrome or people delusional about how their body really is.
I actually don't see that "cogito ergo sum" and Heidegger's "thrown-ness" are so contradictory. Descarte wanted to find a claim irrefutable by logic on which to found his subsequent thought. And he did. The fact he went on from there to separate mind and body in a way that Heidegger's thrownn-ness doesn't require doesn't mean the ideas aren't compatible.
The problem with Descarte's thought is not the first but second therefore: I think therefore I am therefore my mind is separate from my body.