I am (much) less than optimistic about the We as the winning vote, so if you think something different feel free to ignore these lines as the doomsaying of a killjoy.
Now, with the disclaimer out of the way, considering the other stuff that Mathilde has done in Karak Eight Peaks and the extra dose of spooky that the We are going to represent for many... The real world analogy that comes to mind is a nuclear power plant. Something that is undeniably important and often valued, but rarely embraced by the local community, so as far as the locals are concerned I see it ending as more of Mathilde's wizard stuff that they don't really interact much with.
I don't expect things to be much better in regards to visitors, unless the library manages to amass a truly formidable collection. There is an extremely strong in-universe bias against both "wizard" and "weird", so reputation was always likely to be something of an uphill battle, even before spider librarians, and location is sufficiently remote that it doesn't help.
I am also concerned about the ongoing kerfuffle involving the silk trade reflecting poorly on the library. Last I checked, the relevant dwarven guild was making slanderous accusations aimed at We silk and I imagine that any trade partners they may have in the human realms (Marienburg?) are likely to follow if the silk situation ever becomes viable in K8P.
In the plus side, if We silk eventually becomes established as a luxury good is likely to be good for their reputation (in no small part because there will be people invested in helping build that reputation to increase their earnings), which may eventually be good for the library's own reputation, although I expect this to be a long term effect after a fair few years, at least.
On your first point.
I am going to disagree on locals not embracing the library.
It will take time, years, if not decades, but the size and uniqueness is going to be working in the library's behalf, eventually the locals are going to be proud of having it, and not giving a fuck about the librarian while visitors freak out, think Altdorf and College's of magic as an example.
They might not visit the library much, but that is mostly out of not having any reason to.
And as i have previously mentioned, Library being a major economic actor is going to also act in its favour long term.
It is not going to be as central part, no, but then that was a known issue from the vote descriptions already.
Library will amass a truly formidable collection, the question is not if, it is when, and K8P is so out of the way that very few people are going to make the trip no matter who the librarian is, and if the issue is important enough to do the trip, some giant spiders are unlikely to stop them.
Not sure why you think the We Silk kerfuffle is relevant to the library. Though as you mentioned, if/when the silk production kicks in, the sheer economic factors are going to largely negate any problems, even assuming some dwarven guilds are going to actually try to attack a hold as important as K8P.