Dear Armin,
Thank you for reading my essay and for the comments. You raised interesting points, and I will address them.
1. As you know, in mathematics the term "exists" is the same as in logic. For example, "there exists a field which extends the field of real numbers and is algebraically closed". This doesn't have the meaning of physical existence, but rather of logical consistency. Whenever I used "exists", the sense of mathematical existence or physical existence follow from the context, otherwise, I specified that I was talking about mathematical existence or physical existence. To change the terminology would be unnecessary and would introduce confusion. You are right that they can be identified in the context of MUH. You are, of course, right, that the standard terminology is not the most fortunate, but I hoped that my precautions were enough.
2. Probably the reason why you feel that there is more to existence than relations is that one considers more existent the things with which we have a relation, directly or indirectly. But this is relation too. If you could be more specific about that property that escapes the relational viewpoint, that would be helpful. Otherwise, I think my statement is not merely a way out, but is the only way which avoids reference to things which don't have observable effects by themselves.
3. I agree with you about the necessity to avoid anthropocentrism. This is why I wrote "Being able to guess them and then test them would mean either that we are that lucky, or that the universe wants to be completely understood by us, who are just tiny waves on its surface." As you could see, I did not claim that this theory must exist with certainty, and certainly didn't claim that, even if it exists, we can find it. But I think that it is likely that it exists, and even that we find it, simply because we are so close. Most of the physical laws are contained already in general relativity, quantum theory, and the standard model, which really are a small set of laws accessible to us. Of course, this doesn't ensure us that TOE exists and can be found. You also said "To give a specific example, who is to say that it is not possible to formulate a theory of nature from the frame of reference of an observer associated with null geodesics in spacetime?" Well, you are right, physical laws can be described very well in coordinates whose constant surfaces are lightcones. Also, in Finkelstein coordinates and Kruskal-Szekeres coordinates, and also in Penrose-Newman formalism and Penrose-Carter diagrams, null coordinates are used. There is no need for an observer to travel at the speed of light, this is simply the diffeomorphism invariance of laws in general relativity. So the problem of antrhopocentrism introduced by reference frames was solved in general relativity, for other reasons. I think this was a good point you raised, because it could lead to the diffeomorphism invariance, or at least is another good reason to use them.
Thank you for the excellent points you raised, they allowed me to clarify some perhaps unclear elements, and to see some things I knew in a different light.
Best wishes,
Cristi