I don't know how different this is for public vs. academic libraries, but we get 35% of our traffic from search. So being searchable on the open web is kind of critical.
If you're not in a position to get these numbers for your own site, you could check with some peers at other institutions and find out what percentage of their traffic comes through search.
Sonoma County Library
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707 545-0831 x1581
From: Code for Libraries [mailto:[log in to unmask]] On Behalf Of Lauren Magnuson
Sent: Tuesday, December 17, 2013 12:41 PM
To: [log in to unmask]
Subject: Re: [CODE4LIB] Academic Library Website Question
Agree - you'd still need evidence to convince the powers that be. To get
some data to that end, maybe some snapshot usability observation where you
sit down a handful of users at a computer (better yet, on a tablet/mobile
device) and ask them to find the library's website (ensuring that the
browser doesn't default to university.edu). Observe their strategy - do
they go to university.edu and click around, or do they Google? Or do they
use the university.edu search box (which is often a Google site search,
playing by the same rules)?
If your users are Googling, having that data in hand might provide an
argument that your page should be Google-able and you should have some more
visibility/control. Or you could set up some dummy sites and Google bomb
yourself :). (Joking. Kind of.)