I agree with everything Martin said below, but I want to go a step further and argue that judgmental equality is potentially holding us back, in terms of meta-theoretical results AND usability of the type theory.
I can understand that, for programming, we want built-in computation. However, for a foundational system, I think we should strive for something as minimalistic as possible without arbitrary additional features.
It might be a bit controversial to call judgmental equality an "arbitrary additional feature", but there is certainly some arbitrariness involved (why else would Sigma have judgmental eta in some systems and not in others?). It seems to me that the cleanest, most minimalistic, and most canonical choice would be a type theory without judgmental equalities *at all*. This would also be much more in line with the approach of category theory, and I expect that it could make the lives of those of us working with models easier. Is there a deep theoretical reason for the necessity of judgmental equality?
Of course, we want convenience and usability. I just fail to see why this means that judgmental equality has to be a built-in thing (one could say: why does it have to be part of the core). It's possible that, if we do *not* insist on built-in judgmental equality, we could even get *more* convenience. What I mean is this: Let's write WTT for a "weak type theory" without any judgmental equalities (since we're doing HoTT, maybe with some axioms such as univalence, but let's not specify this here). Let's say we know that a certain set A of judgmental equalities does not change WTT, i.e. that (WTT+A) is a conservative extension of WTT (conservative in a constructive sense). We could now have a proof assistant which uses WTT as core but which allows us to "load" the conservativity proof of A. Then, at the beginning of a module, a user can tell the proof assistant that in this module, they pretend that the type theory is (WTT+A); i.e. for the user, it looks as if they are working in (WTT+A) but, in the background, the proof assistant translates everything to WTT. In another module, the user can pretend they are working in (WTT+B) instead, if (WTT+B) is conservative over WTT as well. The idea here is that some proofs benefit more from one set of judgmental equalities, while other proofs benefit more from others. Being able to choose which equalities one wants could lead to increased convenience. (I think WTT+A+B is not necessarily conservative over WTT, so we cannot just always assume all equalities for maximum convenience.)
For example, let's consider a cubical type theory. I'm aware that we don't really know many conservativity properties of cubical type theories yet, but I would be interested in knowing some. Maybe some cubical type theory gives us a set C which our WTT-based proof assistant can load. In other words, instead of saying "hey, we have found a new cubical type theory with amazing computational behaviour, an implementation is on GitHub", people could say "hey, we have found a new set of judgmental equalities, on GitHub is the conservativity proof that you can load into your proof assistant to use it".
I don't think this is something that we will manage to get anytime soon, and I find it very difficult in general to find conservativity results, but is there a theoretical reason which makes this impossible?
Nicolai