If this is something you want to make a difference in, one avenue of attack would be to look into Lamport's "Structured Proof" idea. You are young, and you have come of age at a time that hypertext is readily available and widely used as a medium of academic communication.
Your question prompted me to ask this one, which resulted in me finding this reference after having lost it for 20 years. (So, thanks :). If this is something that you particularly want to dedicate time to, Lamport has laid out a path for you. You could
Do all your proofs in the structured format that Lamport has laid out. He references using HTML for this many times, but I don't know if he has done a proof-of-concept. If not, that could be a contribution you make.
When you come up to the point that you are able to do so, you can attempt to do your own study of existing literature, and find out whether his initial one-third number is actually valid. I don't know of anyone that has tried to validate that. (I don't think you would have much luck at this until you have a field that you know well enough to be able to understand existing proofs well.)
See also this redux by Lamport where he gives the same talk 20 years later or so, and also you can search around in his list of publications for references to "101" to see his various mentions of the structured proof concept as he discusses his other work.
If you start now, and you come up with a good way to do a "progressive disclosure"-type interface to a structured proof, it's possible that you could get good enough at doing proofs this way that it becomes second nature, and may actually be faster (it seems like it will almost definitely be more accurate). I'm saying "faster" is a possibility because it might allow you to clear the noise in your head about a sub-part of the proof, knowing that you can just come back to that bit of it. It could make it easier to think about the whole thing. All the speculative statements in this answer are just that--speculation, I haven't tried any of this myself.
I feel like this is a humanly possible effort, and if this question about the soundness of mathematics is something you are highly interested in (I phrase it that way because it will undoubtedly be a lot of work, and an uphill battle against culture and norms), this is potentially a way you could do something about it.