In spite of being a simple words game, that would make me feel less uncomfortable, as there could be a mysterious field still less mysterious than matter that passes through itself etc. Moreover, couldn't be that there exist some high order terms that play a role only when mass reach big values, i. e. those comparable to a galactic mass? How/why this cannot be the case in general relativity?
I don't think the title of your question and your question are really in sync here and I'm not sure I will correctly answer your question. However, I will give it a try.
To give you a hint of the conclusion. My answer would be, "NO".
There are several clues / reasons why I come to this answer.
- Gravity is not necessarily a real thing. When you read the literature concerning general relativity about gravity you will learn that acceleration can be experienced as gravity and in the fabric of space/time gravity is "merely" a curvature in this fabric that. This means that gravity is highly debatable as something that could be seen as an independent force / object in the universe.
- When gravity exists as a force, whether it is an exchange of gravitons between two objects with a mass, or whether it is a curvature in space / time due to the mass of an object, the phenomenon "gravity" is always related to the mass of an object and doesn't stand on its own as an independent force of nature.
- Dark matter is not a "mysterious" thing that passes through itself. Dark matter is just the container concept of matter out there that doesn't interact a lot with other matter, but just leaves its marks in the universe by showing a gravitational force as interaction with other matter. Since we have only little understanding of gravity and we still have difficulty measuring gravity as a force in the universe, we have difficulty understanding and measuring dark matter directly. This doesn't mean it is not out there.
So, to get back to your second part of the question, "massless gravity", this is not a viable alternative. Gravity exists either by massive objects exchanging gravitons and therefore exercizing gravity upon each other, or a massive object "denting" the fabric of space / time and therefore creating a gravity field. In either case gravity cannot be seen as a farce that exists without the existence of mass.
Secondly, to the first part of your question. Whether we can observe objects directly or indirectly, this doesn't make it more or less realistic. Dark matter is generally accepted to be there. We just haven't found a way to measure it. We can only see that the visible matter in the universe can't explain the gravitational pull on the visible matter and therefore we assume there is some other matter out there we haven't been able to "see". Hence, dark matter.
However, dark matter does interact through gravity and our current successes in measuring gravitational waves might also result in a better understanding of gravity and how to measure the effect of dark matter on the gravitational pull in the universe. So, to answer your question: "No, we call dark matter because we can only observe it through its influence on other matter due to gravity. There is no such thing as gravity without mass."