Why is Wikipedia so bad to use as a source but a random website is not?

A "random" website is just as bad a source as Wikipedia, but for different reasons.

Wikis are bad resources because, in and of themselves, they're not a source, they're a collection of sources that have been assembled by someone else. Aside from having someone else do the work for you, there's nothing to say that the person didn't just make up what's written there. Encyclopedias (printed ones) are just as bad as primary resources, for the same reasons.

Citations from Wiki articles can be good sources, however (but not always). This is because the source itself is likely written, posted, and hosted by an organization that peer reviews or is an expert in said topic. Experts write articles, which get published and reviewed by other experts -- making them legitimate sources for info. These are not the same as "random" websites.

TLDR: Wikis are collections of sources, not primary sources. Primary sources are peer reviewed and published as such.
why is a random website's word any more valid than the person who edited a Wikipedia article.

It's not the fact that something is on the internet that makes it unreliable. It's the nature of the site in question. Some schmuck's MySpace page? Worthless. A paper on a .edu domain? Probably a cite-worthy academic source.
You would think that because wiki has more web traffic that mistakes are more likely to be corrected.

No, you wouldn't. And even if that were true, it's not really the point. There are several things which make wikipedia immensely problematic as source material.

First, it's not a fixed source, i.e., the page that the reader sees when they look up your citations may or may not be the same as the page that you saw when you wrote it. True, one can check the change log, but that's a huge pain in the ass and not necessarily helpful anyway. How can anyone check your work if there's no way of verifying the sources you used?

Second, it's mostly anonymous. We have no idea who is writing this stuff most of the time. That's important, because if the person doing the writing has a vested interest in the issue, we need to handicap their opinions appropriately.

Third, citation on wikipedia is ridiculously inadequate. Even when citations are supplied, there's almost never any indication that the article is engaged with the conversation in the literature as a whole. As long as there is anysource anywhere which says what the wikipedia editor wants it to say, it goes in. Never mind that the source in question might be outdated, mistaken, a fringe view, unreliable, discredited, whatever. There's no methodology for evaluating sources, so the resulting articles vary pretty drastically in quality.

Any other website which has those problems should not be used as a source for academic work. Any website which does not have those problems is theoretically fine. If you were going to cite to a newspaper, there's no reason you couldn't cite to the online version. Same goes for academic papers, books, you name it. But there are inherent problems with wikipedia that make it entirely useless for academic work.