It's definitely not mainstream, and wouldn't be mainstream even if it was the work of someone at Oxford or Harvard.
In the world of quantum gravity, there are frameworks that preserve space as a continuum, like string theory and various field theories, and then there are more combinatorial theories like loop quantum gravity and causal dynamical triangulation.
From what I understand so far, "Emergence Theory" combines a combinatorial idea of quantum gravity, to be based on tetrahedra, E8, and the golden ratio, with a metaphysics in which "measurement" is happening everywhere all the time. It sounds like panpsychism (everything is conscious), or perhaps what David Chalmers calls panprotopsychism (everything is almost conscious).
Occasionally university physicists publish comparable ideas (e.g. Teruaki Nakagomi's "Quantum monadology", or maybe some of David Finkelstein's later work), but those are eccentric works. Also, if we focus just on the geometric aspect of Emergence Theory... so far, it seems to have even less substance than
Louis Crane's E6 neutrinos. Is there a lagrangian or a partition function somewhere, or is there just an aspiration to build a model out of the ingredients specified?
So I would think of it like this. Just with respect to quantum gravity, the most mainstream you can get is the quantum field theory of gravity you get by quantizing general relativity in various straightforward ways, like perturbative field theory or Hawking's wavefunction of the universe. Then you have the unproven theories which try to build on that in some way, above all string theory (for numerous reasons) and asymptotic safety (because it does have a few successes). Then you have the combinatorial frameworks, above all loop quantum gravity. Most of us in the continuum camp have strong technical criticisms of loop quantum gravity, but I think I have to count it as mainstream.
The purely geometric part of Emergence Theory appears to be two steps removed from mainstream combinatorial quantum gravity, if we rank theories by mathematical substance. One step removed would be Louis Crane's model, linked above, mostly because it hasn't been shown that it makes sense mathematically. Crane is a mathematician, and he has some ideas, but he hasn't defined a model even at the level of the other papers he's written on the subject. And then "Emergence Theory" doesn't even have something like Crane's E6/A4 duality. Though maybe it hopes to employ something from Lisi's E8 theory??
Anyway, what I'm saying is that mathematically, unless I've missed something substantial, there doesn't seem to be much there in "Emergence Theory" except the hope that the prioritized ingredients can actually be soldered together into a functioning theory. So the "Research Group" is a group of people - they are actual physicists - tinkering with these concepts in the hope of eventually having them cohere. And they are writing and even publishing some papers, in journals of mixed quality, about these other investigations. For example, in their publication list I saw a paper about cosmology, and another one about quasicrystals. What I believe they're doing is tweaking established models in ways that they hope would eventually be motivated by Emergence Theory, once it's a real theory. But in those cases, for now all they have is the modified establishment model.
On the basis of what I've seen, my instant opinion is that there's too little deep math for Emergence Theory as it is to be the answer to anything, and that its apparent adherence to the combinatorial philosophy of quantum gravity also ought to doom it. But I do find the metaphysics intriguing, and even the idea that some of these remarkable algebras like E8 might have a "cognitive" interpretation (see the 1990s collaboration on octonions, between Kent Palmer, Onar Aam, and Ben Goertzel). E8 does have a self-referential quality to it. So I think their concrete research output is second- or third-tier, it's more contributions to the ever-growing literature of models that aren't right and which have some flaws of principle. But I like some of their spirit.
But that's just a personal opinion, and one based on a rather quick and superficial examination of their material. Anyway, by any reasonable standard, they are definitely not mainstream.