>academia or traditionally prestigious sources.
You can't really put ArXiV and "traditionally prestigious sources" in the same sentence. Any crackpot can (and does) post on ArXiV.
The reason why I'd be more comfortable citing a paper on ArXiV than a random web page (as a mathematician):
1. ArXiV is the archive for papers. You put something there, it's not going away. You can't even remove your own paper once you make it public there[1]. It's a reliable place I can refer people to.
Personal websites? Well, my own very reliable academic web page has been removed by sysadmins without notice after graduation because "we don't maintain accounts".
2. Unrelated content. ArXiV won't have it - unlike, say, the 4chan thread in question. Do I really want to explain to people that I wasn't citing getting "14 * 14! derps out of the way"?. Not to mention ads, memes and NSFW content - which is also there[2][3]
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The other reasons are less important, but are still valid:
3. Presentation. Something on the ArXiV probably is formatted and structured like a readable paper; there are baseline expectations. Reading someone's scratchwork is borderline painful, and a proof on a webpage is usually just that.
Compare [3] to a rewrite[4] - which one is easier to understand at a glance?
4. Reputation. This is different from "traditional prestige". The reputation of ArXiV is that most of the people who do serious work put it there. Most of the people produce nonsense (possibly including yours truly) also put it there, but the probability of any work having been done by someone who is willing to put effort into it goes down dramatically if the work is not on ArXiV.
5. Signaling. Since it's so easy to post there, any work not ArXiV is signaling "I don't care about my work being available to other people in the field". Why one read something that the author doesn't want them to? If they did, they'd put it on the ArXiV.
6. Attribution. A result on ArXiV makes it easy for me to attribute the result to someone. A thread on 4chan? Do I attribute it to everyone who participated? A single anonymous author?
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In short: people give credit to blackboard scribbles and napkin proofs that came out of discussions at conferences, but you won't see "4th beer coaster, such-and-such conference banquet, 2018" as something that people cite.
A proof on 4chan is somewhere in the same category.
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The disclaimer here is that this applies mostly to new results in mathematics. Old results are not necessarily available online, and often we don't have a choice. Practices in other fields might be different.
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[1] https://arxiv.org/help/withdraw
[2]https://warosu.org/sci/thread/S3751105#p3751197
[3]http://4watch.org/superstring/
[4]https://docs.google.com/viewer?a=v&pid=forums&srcid=MTUwMTUx...