I am a mechanical engineer with a love for theoretical physics. I agree with your comment. Sometimes I feel are stuck thinking in the dimensions we can perceive. Which is why we call Dark Matter "Dark" - we can't perceive it with the way we currently think. Although we should not think in 2D to solve a 4D problem, it may actually help us visualize it. I will explain in a minute. As an "outsider looking in", what I can see is that Dark Matter really is all about 4D - and even higher D's. I've read in more than 1 occasion that our Universe can allow up to 11 dimensions. I believe we all can agree that the presence of Mass on the fabric of space-time is what causes gravity - and that space-time bends. Therefore, there has to be more "room" to allow for the bending.
In one of my Materials Engineering courses, I remember the term "Elastic Limit". In essence, when a material is stressed to the point of exceeding its elastic limit, there will be a permanent deformation. For example, if we had a sheet of a rubber-band like material stretched out and put a lead ball in the middle, it will deflect. Once the ball is removed, it will return to it's original shape. Now, let's imagine something like a flat sheet of Aluminum foil. Imagine if we put that same lead ball. If the sheet is stretched taught, not only will it bend/deflect, there may be a permanent deformation. This may not be the best example, but we can visualize this. So - what is left behind? A sheet with a curved, dimple on it. There is nothing there, but we would see and feel the consequence of that deflection if we where " tiny 2D beings" living on that sheet.
Therefore, what if what we are perceiving as Dark Matter really is only a "permanent deformation" of the fabric of Space-Time? Could it be that something extremely massive existed there at some point - and moved on?
Maybe in a "previous life" of our universe, there was something massive there and left its "impression". If we manage to detect what is really happening to space-time at those points, we will have an idea of what Dark Energy is as well. After all, the energy-matter has to balance.