Easy proof of a big conjecture based on possibly faulty paper – what to do?
It seems that you have found an easy one-page proof of a very interesting result along the following lines:


  Theorem 1. The results of Papers A, B, C imply Conjecture BIG. Therefore if those papers are correct, then Conjecture BIG is true.


(In the rest of the answer I will assume that your proof is correct since you said it is short and was looked at by experts, but to the extent that you still have doubts, it's obviously advisable to very thoroughly vet the correctness of the proof by going over it yourself and showing it to more people with the relevant expertise.)

Now, normally one would go directly from here to the statement that you've proved Conjecture BIG, but the twist here is that Paper A is complicated and has missing details (hmm, never seen that before...) so you are doubting whether it is correct and are reluctant to declare yourself to have proved Conjecture BIG. However, it's worth pointing out that there's already quite some cause for excitement, since you have already proved the very interesting fact that


  Theorem 2. Either Paper A is incorrect, or Conjecture BIG is true.


Given that Conjecture BIG is, well, big, and Paper A is a published paper, it seems almost certain that this is already a publishable result, and in any event (even if you don't care about publishing) one that the scientific community will be very interested in. Precisely how exciting it is depends of course on whether you can go even further and figure out which of the two scenarios applies, and if so, what you discover, and, in the event that you discover that it is the first scenario (Paper A is incorrect), whether you are able to correct the results of the paper by completing the missing details or by correcting whatever errors it contains.



Now let's discuss what you should do. I see three main options for how to proceed:


You can write up Theorem 2 and its easy proof (along with an explanation of why you are formulating it in such a way rather than simply saying you've proved Conjecture BIG), put it on the arXiv, then submit it to a journal, and let others take it from there.
You can wait before publishing your findings and first of all spend time studying Paper A so that you understand it well enough to determine whether it is correct. While doing so you will also attempt to fill in any missing details. Once you have determined whether the paper is correct, and (if it's not correct) either managed to correct it (by filling in the gaps and/or correcting erroneous reasoning) or failed to correct it, or convinced yourself (e.g. by finding a counterexample) that its results, not just its proofs, are actually false, you will then be in a position to write a paper that contains a proof of one of the following results:



  Theorem 2 (same formulation as above - in the event that your study of Paper A didn't lead you to any meaningful conclusion)


  Theorem 3. One of the results of Paper A is false (because its proof is incorrect, and you have proved that the result is also incorrect).


  Theorem 3'. Conjecture BIG is true (because you have ascertained that Paper A is correct and your initial doubts were misplaced).


  Theorem 3''. Conjecture BIG is true (because Paper A is essentially correct, but has some missing details you've managed to fill in).


  Theorem 3'''. Conjecture BIG is true (because the result of Paper A is correct, although the proof in the original paper has an error in it, which you have managed to correct).


A variant of option 2 above is to recruit professional help in studying Paper A, either privately by approaching someone you know who may be interested in becoming a collaborator, or publicly, for example by posting your thoughts on the paper on MathOverflow.


As for which of these options is the best one, that depends on how difficult it would be for you to try to pursue options 2 and 3, and whether you want to maximize the amount of fame/glory/credit you get from this discovery. The point is that each of Theorems 3, 3', 3'', and 3''' would be a stronger and more interesting result than theorem 2, so if you prove them your discovery would make a bigger splash, and conversely, if you leave them for someone else you are giving up a potentially very useful opportunity for professional credit and possibly (if this interests you) career advancement.

Thus, if you want to publicize your discovery with the least amount of effort and in the least amount of time and don't care about maximizing the credit you get from it, you should pursue option 1. If you are willing to take the time and trouble to look more in depth into Paper A, you should pursue options 2 or 3, with option 3 being a better approach if Paper A is technically too complicated for you to study by yourself. One additional concern in that case is the issue of priority, so if you are worried that someone else may discover your one-page proof while you are busy studying Paper A, you should make sure to document your discovery including relevant dates, show it to a few trusted people, and limit the time you are working on this to a reasonable amount of time (say, no more than 3–6 months).

I should add finally that you have no ethical obligation to determine whether the results of Paper A are correct (emailing the author to ask them their thoughts is certainly a nice thing to do, but you said in the comments that you tried that already and it hasn't led you anywhere). So from an ethical point of view, any of the options I outlined above is completely acceptable.