Heidegger is slippery and challenging. Unfortunately, my experience of Heidegger is limited but I will try to clear up here as much as I can. I think it is a often the case that Being and Time is read as text that both refutes and cannot escape from the tyranny of the Cogito's metaphysics. This is partly because it is difficult to understand that Heidegger's concept of being-in-the-world replaces the dualistic ontology of the Cogito. But I think that this confusion is largely due to the Cartesian ontological assumptions which are entrenched in our philosophical intuitions about consciousness, making it difficult to approach Heidegger. With that said, before I dive into Heidegger, I'd like to make some remarks about intentionality and the Cogito. Many interpretations of the Cogito state that it is intrinsically structured around intentionality; consciousness can only be consciousness if thoughts are about something, even if they are about themselves. So, Descartes argues, the inward directedness of mental life is maintained in absence of any extended object(s) which can be consciously obtained. Thus, the dualistic, metaphysical conclusion that the mental can exist without the material, the subject without the object. Note, that in order to count as "I," or a mind thoughts must at least be capable of representing something, namely itself. Thus, for Descartes, minds are ontologically self-sufficient because thoughts are ontologically self sufficient, so you are right to question that the represented thing needs to designate a mental entity or property, or that it needs to be anything at all. It is here that Heidegger does something really exciting.
Being, for Heideggar is not just anthropology We cannot strip the world of all knowable things without stripping away the knowing engine -- the subject. Heidegger, as you point out, repudiates this conclusion. Yet, Being and Time seems haunted by For him, Being is a question of minds embodied and embedded in beings and their world.