When do you get to the point of writing proofs that need to be so complicated that verifying the details becomes a great burden on others? Keyword here is need, many people thought I meant this intentionally rather than as an inevitability. I recently reread the article on Fukaya's work at Quanta and looked into the situation a bit more myself. Throughout undergrad and just starting grad school, I've never had to create a proof so complicated that it takes tons of corrections in order to know it's solid. Nobody writes down to the axioms and everyone leaves out details at some level, but I've never skipped from one step to another without knowing how to proceed. I would always make sure it follows from a result I already know, and even then, usually the justification with applying it is at most 1-2 sentences.
Due to a separate discussion earlier this year, I compiled a list of the longest or most complex proofs (as of June 2022) I've made myself. This excludes any solutions I gave when practicing or taking math competitions, any tests I took on paper, and any questions I answered online. Many of these should be added to the list, but it would be a pain to go through all my old papers and take pictures of every long solution. I also have some research this summer, but the proofs aren't elaborate. In any case, you may judge the statements in the preceding paragraph yourself by looking at the Dropbox.
What concerns me is that even in the future, I can't imagine publishing papers so complex that they have dozens of sentences which require a page or more of justification to establish from the previous sentence for someone who doesn't already know the standard argument. Yet in order to achieve groundbreaking work in the way mathematicians working in fields which take a lot of prerequisite building to start accomplishing stuff such as algebraic geometry or modern number theory do, you need to be able to construct such proofs. As it happens, I'm interested in those fields. So how do you arrive at the point of making one of these large papers which takes others years to fill in details and yourself years to publish clarifications giving all the elaborations?
 A: If your goal is to make arguments that need much more justification than you've given them, you're off to a great start with your argument that, because Fukaya wrote a paper that skipped a lot of details, everyone needs to write such papers to do groundbreaking work in algebraic geometry or number theory. That seems like a lot of generalizing from one example.
Plenty of mathematicians manage to do groundbreaking work while writing very clearly. In algebraic geometry and modern number theory, maybe the best example is Serre. There's also great work done with relatively short papers.
But, that said, I think you will, if you continue doing math, write some very complicated arguments. This will be easier than you think! When I first started math, I had no idea how people wrote such long papers or even books focused around a single argument. But now I have written multiple long papers. How did I do it? Every time, I tried to write a short paper and failed. At some point, it will become that simple.
Less glibly, the way too come up with a complicated argument is to start with a hard problem (a lot harder than problems on exams, in math competitions, or most questions asked online), and think about it for a long time. At some point you'll come up with a strategy, and that strategy will break the problem down into several subproblems. You might need to think about these for a while, and each one will become a series of further subproblems. At some point you'll get problems that don't feel that different from questions you've been asked in exams, competitions, or online. You'll solve these, and start writing it up, and eventually obtain something that is almost certainly much longer than you anticipated upon starting.
Note in particular that the specific list of proofs you're comparing to are largely problems from  math competitions and tests. Questions on math competitions and tests are designed, among other things, to lend themselves to short answers with each reasoning step unambiguous. Imagine having to grade or score answers that were written like sketchy research papers! The fact that you wrote proofs that are easy to understand on questions that were specifically designed to have easy-to-understand proofs is maybe not so indicative of the limits of your proof-writing capabilities.
Finally, read there's more to mathematics than rigour and proofs. You may not have reached the "post-rigorous" stage in very many fields of mathematics yet. The ability to toss off "oh by standard arguments we can prove ..." without actually writing down the arguments and, when the details are eventually filled in, be correct about what can be proved is characteristic of the post-rigorous stage. You get to the post-rigorous stage by continuing to work on problems in a particular field, thinking deeply about them, reading papers, speaking to experts, attending seminar talks, and so on.
