How can we expect that increasingly complicated proofs are replicated when so few people can understand them in the first place?
My answer to that is that we do not expect them to be replicated in the usual sense of this word (repeated and included into textbooks with just minor cosmetic and stylistic changes). Rather we expect them to be gradually simplified and streamlined either through changing the proofs themselves by finding a shortcut or replacing the whole argument with a completely different one, or by building a theory that is locally trivial but proceeds in the direction of making the proof understandable and verifiable much faster than the currently existing one. The latter is exactly what Mochizuki tried to do though his goal was rather to just reduce the difficulty from "totally impossible" to "barely feasible" and the prevailing opinion is that he failed in the case of the ABC conjecture though he has succeeded in several other problems.
The first approach is more common in analysis (broadly understood), the second is more common in algebra (also broadly understood), but you can try to play either game in either field. My own perception of what is proved and what is not borders on solipsism: I accept the fact as proven if I've read and understood the whole argument or figured it out myself. So most mathematics remains "unproved" to me and, apparently, will stay unproved for the rest of my life. Of course, it doesn't mean that I'm running around questioning the validity of the corresponding theorems. What it means is that I just never allow myself to rely in my own papers on anything that I haven't fully verified to my satisfaction, try to make my papers as self-contained as possible within practical limits, and that I consider the activity of simplifying the existing proofs as meaningful as solving open questions even in the case when the proofs are reasonably well-known and can already be classified as "accessible". But not everybody works this way. Many people are completely happy to drop a nuke any time they have an opportunity to do it and there is nothing formally wrong with that: the underlying point of view is that our time is short, we have to figure out as many things as possible, and the simplifications, etc. will come later. Probably, we need a mixture of both types to proceed as efficiently as we can.
So I would say that the mathematics is reasonably immune to this crisis in the sense that mathematicians are aware of the associated risks, take them willingly, and try to gradually build the safe ground of general accessibility under everything though the process of this building is always behind the process of the mathematical discovery itself. The same applies to physics and medicine though the gap between the "front line" and the "safe ground" there may be wider. In fact, it applies to any science that deserves to be called by that name. As to the so called "social sciences", they are often done at the level of alchemy and astrology today in my humble opinion (and not only mine: read the Richard Feinman critiques, for example) but we should not forget that those were the precursors to such well-respected sciences as chemistry and astronomy/cosmology, so I view the current crisis there as a part of the normal healthy process of transitioning from the prevailing general "blahblahblah" and weathervane behavior with respect to political winds to something more substantial.
Edit: Paul Siegel has convinced me that things have indeed changed since the time I took (obligatory) courses of Marxist philosophy and the history of communist party, though this change may be not easily visible to the general public because it mainly happens outside academia and is driven primarily by company business interests, so a huge part of it occurs behind closed doors (Paul, please correct me if I misinterpreted what you said in any way). So my statement that the current social sciences are not capable of something beyond general blahblahblah is no longer valid and I retract it. However I still maintain the opinion that it is blahblahblah rather than hard data analysis or other scientific approach that drives many public political and social discussions and decisions of today (I don't know what happens here behind the closed doors, of course, and it may be that, like in advertising, what we see is just what shepherds choose to show to their sheep to drive them in the direction they want, but I prefer to think that it is not exactly the case). If somebody can convincingly challenge that, I would be quite interested.
Apologies to everybody for switching this discussion to a sideline.