But there is one: to save the precious time of valuable people. I think it's an interesting suggestion.

I, too, really like the idea, but I wonder if, in practice, it wouldn't work because companies don't introspect deeply enough to understand what makes them distinct? For instance, if what's remarkable about your company is that you have a great culture where people goof off a lot and have a lot of fun, and still manage to get a passable amount of work done, would anyone dare to self-identify that way?

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.

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.