In terms of scaling, if you're having to screen so many employees that it becomes your bottleneck, then it'd be a business on the path to success. It doesn't need to scale non-linearly, because each candidate screened is worth, on average, some dollar amount.

I see it as an analogy: YC is to Angels & VCs as <this thing> is to Companies that are Hiring. If you get into YC, you are definitely a noteworthy team. Furthermore, YC is able to prune down 1,000+ applicants to under 40 in a short amount of time. They probably turn down 75% of all applicants in 20 minutes or less per applicant. If it's your job to discern, you're going to figure out how to do it efficiently. They also attract very good applicants.

Why can't there be something like this for brilliant people seeking jobs just like YC is for brilliant people seeking to start companies?

You're right, though, about building connections with companies. Though, in this case the "connection" has more to do with reputation of "offers only Google quality candidates, via a rigorous screening process" than "hey, they are hit or miss but have provided some good hires in the past".