The situation with Mochizuki and abc is not a good one, but I think it is a mistake to regard it as an example of a "replication crisis."
First of all, if you read about the work of Yitang Zhang (e.g., the New Yorker article), you will see that you don't have to be a "celebrity" to make a major breakthrough and have it be accepted.
Secondly, at this stage, the primary problem with Mochizuki's proof isn't that it's enormously long and complicated. Yes, that was a stumbling block initially. But people did eventually take the time to study it. Several people independently ran into a problem at exactly the same spot in the argument, namely the notorious Corollary 3.12. The proof of this corollary is unclear to a lot of people. Now, what usually happens when a mathematician presents a new proof and people can't follow it is that the mathematician provides additional details. To a non-expert, this might sound strange; if the proof is correct, shouldn't all the details already be there in the first place? However, professional mathematical papers are written in a style that assumes that the reader has a certain amount of expertise and is able to fill in "straightforward" intermediate steps. This is usually a good thing because it draws attention to the crucial points in the argument, without drowning it in an unnecessary ocean of details. However, there is always the risk that the reader will have trouble fleshing out the intermediate steps. But it is generally understood that if a reader has such trouble, then the author of the proof will be able to elaborate and supply more explanation until the difficulty is cleared up.
Sometimes readers have trouble because a long calculation has been elided. But that is not the case with the proof of Corollary 3.12. The problem is a conceptual one; the terminology and the intended flow of the argument is not clear. One of the main contributions of Scholze and Stix has been to phrase the difficulty in a vivid manner. Roughly speaking, they say something like this, "It seems that this type of argument can't work. How do you intend the argument to go? If it's supposed to go this way, then here's the problem you'll run into. On the other hand, if it's supposed to go this other way, then you'll run into another problem. Which is it? Or is it some other argument? How do you get around the obstacles we've pointed out?" By spelling it out this way, they have enabled a lot of other mathematicians, who aren't necessarily experts in the whole proof, to understand what the sticking point is, at least to some extent.
When this kind of question is raised, one hopes that the response will be something like this, "Ah, I see your confusion. The argument doesn't proceed along those lines. It proceeds along the following lines. When I said X, what I meant was such-and-such. The obstacle you cited isn't a problem because of such-and-such. Here's a simple example to illustrate what I mean." But this is not how Mochizuki responded. For example, in his
response, on page 43, he wrote that "my oral explanations, over the past few months, to various colleagues...of the misunderstandings summarized in §17 were met with a remarkably unanimous response of utter astonishment and even disbelief (at times accompanied by bouts of laughter!) that such manifestly erroneous misunderstandings could have occurred." This is outright ridicule of a serious question. By contrast, neither Scholze nor Stix has said anything remotely insulting or personally offensive in this entire affair. It is not just Scholze and Stix, but a large community of mathematicians around the world, that can read Mochizuki's response and see that Mochizuki has not addressed the very specific gap in the proof that has been highlighted. Even those who want to give Mochizuki the benefit of the doubt (and I have met some such mathematicians) will quickly acknowledge that a major objection has been raised that should be answerable if the proof is correct, but that has not in fact been answered.
It is not just a matter of a personal falling out between Mochizuki and Scholze/Stix. Several people, such as Fesenko and Yamashita, claim to understand the proof. There are others such as Taylor Dupuy who don't understand the proof of Corollary 3.12 but are willing to listen and put in the effort. If the proof is correct, then it should be possible for such folks to get together and straighten everything out. That hasn't happened. The circumstantial evidence is therefore strong that the gap is real.
In short, while it's theoretically possible that mathematics could reach a point where the trouble is that it's so complicated that competent and willing people are unable to explain their ideas to each other, this is not an example of that. It's most likely a case where there isn't a valid proof at all. Even in the unlikely case that the proof is valid, the trouble is still not the complexity of the proof, but rather the sociological rift that has occurred.