I think accuracy and intellectual rigor should always be the primary concern when citing a source. Accessibility is nice, but it should not be a policy. Sure, not everyone can access various obscure texts, but then again, if you live in certain countries, you might not be able to access many websites either. Given the choice between an out-of-print academic history text (which may be practically inaccessible for some users) and a sleazy website which doesn't cite any reliable sources and just spreads famous apocryphal stories with no fact-checking, I think we'd all agree that the first is a reasonably good source even if you can't read it personally, and the second is no better than having no source at all.
On the other hand, a self-published book making clearly absurd claims is less reliable than a well-referenced biography on a university history of science department website, so books aren't universally more trustworthy; it's a question of the author's methodology and references. Primary sources are the best insofar as they aren't biased. Secondary sources citing primary sources are typically trustworthy if they have good methodology. Even tertiary sources and higher are sometimes going to be good enough for relatively uncontroversial claims, but if a source is making claims without citation or with very weak citation you should be skeptical whether it's a website or a book. The average book is probably better than the average website, but there's a great deal of variation for both.
That said, it's important that the answer on our site is complete without having to consult any external source (be it a link or a book). If a question is raised about the actual content of the source, it seems reasonable to ask the person citing it to include a direct quote. The poster is probably not the only person with access to the content. Many users here are likely to have academic positions, either in math/science or in history, and as such have access to a great deal of relatively rare books; it would be much worse for this site if we decided these were invalid sources (thereby locking away a great fraction of knowledge on this topic). It's pretty unlikely that someone will find a source that no one else has any access to at all. So at least someone here should be able to verify any seriously controversial citations, but hopefully we can trust each other enough to not just lie about the content of a source that this will only rarely become an issue.
Addendum: I don't think we should put too much concern into questions like "Is source A better than source B?" in general. If both are acceptable sources which make the same claim, include whichever one you personally prefer (or both if you like). If they say different things, and both are similarly credible, rather than trying to sort out the controversy and determine which is correct (which would be original research, not the purpose of this site), you're better off saying that it's a debated point, and that A says X, B says Y, C says Z, etc. So while I support using the best sources available for any given question, if it isn't immediately clear which source is better (i.e. more accurate), don't try to force yourself to decide.