You're so spot on. Some companies do not introspect deeply enough, nor do they have the desire! I've met some of them and guess what –– they don't end up creating a profile on Key Values. Honestly, I think it says a lot about team who put in the time and care to create a profile. The exercise of selecting 8 values and qualifying them signals a real desire to invest in people, culture, and to make sure values actually translate into behaviors.
Re: your example, I see it as my job to encourage them to communicate who they are loudly and proudly! There's pressure on both sides (job-seekers and employers) to check all of the boxes. It doesn't make sense to do that at all though.
I think there are a couple of confounding problems. One is the "Astrology" problem. Well-written horoscopes are written in such a way that people want to believe these things about themselves- and they are nebulous enough that they can. Who doesn't want to think of themselves as being a great friend when someone really needs you? In the case of your website, I think you should consider giving scenarios and force people to make choices between options on a Likert scale.
The other issue is that companies may truly be trying to find the most intelligent (for whatever that word means) person as long as she or he isn't malignant- someone who will be successful in future endeavors after the one they are hiring for is finished. In that case they are just dragging a wide net and trying to get the biggest fish. This is all throw out the window, of course, when some of their screeners have ad-hoc and very personal criteria for rejection (two typos, or not being gender-inclusive, or whatever). Then the process becomes ipso facto a cultural screening process.
Re: your example, I see it as my job to encourage them to communicate who they are loudly and proudly! There's pressure on both sides (job-seekers and employers) to check all of the boxes. It doesn't make sense to do that at all though.