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.