See other comment. It's sound but not decidable.
It tends to be a good idea to give some definition, then prove a bunch of lemmas that characterize the object, and finally forget the definition.
I think one of the reasons is that Lean's approximation of defeq still works well enough in practice. As mentioned by others, you never really run into the kind of counter examples that break theoretical properties, but instead into examples where checking defeq is inacceptably slow, which remains an issue even when defeq is decidable.
From my understanding CiC is used because it allows using a unified language for proofs, specifications, programs and types, which greatly simplifies the system. For example, in Lean 4, more or less the entire system is built on the same core term language: Proofs, propositions / specifications, programs, tactics (i.e. meta programs), the output of tactics and even Lean 4 itself are all written in that same language, even if they don't always use the same method of computation. In my eyes, for the purposes of writing and checking proofs, CiC is quite complex; but if you want to capture all of the above and have them all checked by the same kernel, CiC seems like a fairly minimal metatheory to do so, even with undecidable type checking.
There are a few other examples but this is the one that immediately sprung to mind. The complexity of CiC is very much about decidable typechecking, it's not fundamental to writing a sound dependently typed proof assistant. And anyone who has actually worked on a kernel for CiC knows that while it is "minimal" compared to what people want to do with it, no efficient kernel is nearly as minimal as something like λP2 with dependent intersections.
In terms of stuff that do go more or less directly through the kernel, some followup papers to the one I linked discuss how to translate inductive definitions etc. automatically into a form accepted by the simpler kernel, so I am even more confident that this would not present a problem.
MSR funds a single developer. The Lean community consists of at least 30 or so active contributors right now, with many more having contributed in the past. Most folks work in academia. Much of Lean's success is absolutely due to the work of that single developer, but I don't think that the playing field is uneven due to MSR's funding.
That particular paper is from 2016 and the type theory presented there is still lacking a number of features from CIC, with those particular extensions being listed as further research. Work on Lean started in 2013 and the kernel in particular has not been touched in a long time for good reason. Around that time, CIC was already battle-proven with there being popular and well-tested reference implementations to learn from. Since then, all the fancy work has been done on the system surrounding the kernel.
I believe one of the goals at the start was that the kernel should be sufficiently simple and small so that it is easy to ensure that the kernel is correct. Despite the apparent redundant complexity that you suggest, the implementation still seems to be sufficiently simple to reach this goal, as no proof of false that was a result of a kernel bug has been reported yet. CIC being well-tested surely also contributed to that.
Another reason is that, as I understand it, it was not always obvious that Lean would break decidability of defeq. Afaik proof irrelevance was only baked in with Lean 3, while it was an option before that. By that point I don't see much of a reason to rewrite an entire kernel that has worked fine so far.
I also vaguely remember that there were arguments related to performance in favor of the current system (e.g. versus W-types), but I don't really understand these things well enough to make that point with confidence :)
Here's an analogue. It's like saying "Goedel proved that maths was undecidable therefore maths is unusable and you shouldn't prove theorems". The point is that Goedel's observation is of no practical relevance to a modern number theorist and does not provide any kind of obstruction in practice to the kind of things that number theorists think about.