First of all, I'm very unsurprised that you can get special and general relativity out of something like this. Relativity fundamentally just isn't that complicated and you can see what are basically relativistic phenomenon pop out of all sorts of natural setups where you have some sort of space with an emergent distance metric.
The real question is how this approach handles quantum mechanics. The fact that causal graph updates produce branching structure that's consistent with quantum mechanics is nice—and certainly suggestive that graphs could form a nice underlying substrate for quantum field theory (which isn't really new; I would have told you that before reading this)—but it's not a solution in and of itself. And again what the article calls “branchial space” does look vaguely like what you want out of Hilbert space on top of an underlying graph substrate. And it's certainly nice that it connects entanglement to distance, but again that was already theorized to be true in ER = EPR. Beyond that, though, it doesn't seem to really have all that much additional content—the best steelman I can give is that it's saying “hey, graphs could be a really good underlying substrate for QFT,” which I agree with, but isn't really all that new, and leaves the bulk of the work still undone.
That being said—credit where credit is due—I think this is in fact working on what is imo the “right problem” to be working on if you want to find an actual theory of everything. And it's certainly nice to have more of the math worked out for quantum mechanics on top of graphs. But beyond that I don't think this really amounts to much yet other than being pointed in the right direction (which does make it promising in terms of potentially producing real results eventually, even if doesn't have them right now).
TL;DR: This looks fairly pointed in the right direction to me but not really all that novel.
EDIT 1: If you're interested in some of the existing work on quantum mechanics on top of graphs, Sean Carroll wrote up a pretty accessible explanation of how that could work in this 2016 post (which also does a good job of summarizing what is basically my view on the subject).
EDIT 2: It looks like Scott Aaronson has a proof that a previous version of Wolfram's graph stuff is incompatible with quantum mechanics—if you really want to figure out how legit this stuff is I'd probably recommend taking a look at that and determining whether it still applies to this version.
From the perspective of mathematical logic, string replacement systems could be as powerful as a full functional computer. The proposed graph evolution systems are of the same power too. The author provided many well explained good features of the system and I was persuaded to try to think some science topics from the viewpoint of "graph evolution".
If in future the author or others can obtain new physics findings by using this system, then evidentially the new "fundamental ontology" had some advantages.
However, at this moment, I did not find any provable advantages of this system comparing with other string replacement system yet. I would like to view this work as a thoughtful and well explained scientific investigation ----but the value to people is not proved yet.
Ref: The power of string replacement was well explained in the last article of the book: <The essential Turing>