A question for extreme signalling theorists

I think we all agree that signaling is an important part of the relationship between education and wages, and it's simply an important facet of human behavior to keep in mind. But some people think it's a lot more important than others and - much like voter irrationality in prior years - that share will increase significantly in the next couple years.

So maybe you guys can help me with something I was thinking about.

Some schools have tougher admission standards than others and if you can get in a tough school you try to because the reputation of that school is going to signal something good about you. Going through four years at a tough school shows that you can handle tough situations that are structured and perhaps that have a professional or white collar flavor to them.

But what I don't get is this: academic labor markets, or the market for professors, sorts in the same way. You have to have a PhD from an excellent school to get a job at an excellent school or a very good school. You have to have a PhD from a very good school to get a job at a very good school or a good school. If you get a PhD at a good school life will be harder for you than if you got a PhD at a very good school or an excellent school, and you should probably think of a variety of jobs aside from an academic job. (I am assuming here the traditional assumption that PhD programs are mostly about human capital investments and not signalling - although if you've ever tried to apply for a research grant without a PhD you know it's part signalling too).

This sorting makes sense on the students' end of things, but it doesn't make nearly as much sense (in an extreme sorting scenario) for the academic labor market end. You don't have to have an excellent academic pedigree to make undergrads work extremely hard. I'm not the most brilliant at math - not close. But I know more than enough to make life hell for undergraduates and separate the wheat from the chaff - which is what is allegedly done in the signalling model. A lot of economics departments don't even require that undergrad economics be taught with calculus. Simply making that leap is something anyone with a graduate degree from anywhere in the country could do in the curriculum. In other words, you don't need a top-tier PhD to perform the task of providing signals to employers about students. That's relatively easy. Even bad professors (who are usually bad because they're bad teachers and not because they don't know the material themselves) can rank student qualities pretty well (which is all the signalling model asks of them).

So what explains the high correlation between the toughness of a school for undergraduates and the caliber of the professors they hire? It makes no sense under the signalling model!

If you think of professors as doing two jobs - assigning students signals (i.e. - grades in a course) and research - you would actually expect the brilliant researchers to eschew schools that were known for rigorous assignment of signals. Teaching undergrads is a distraction from research. Let the professors with degrees from third or fourth tier universities challenge the undergrads at the really tough schools like Harvard or Princeton. Leave the professors with degrees from top tier schools to teach at the no-name college where they can get research done (or let them do it at independent research institutes with no signal-assignment obligation at all).

But this isn't what we see. We see the smartest professors teaching at the schools that are the toughest for undergraduates. We also see institutional arrangements that force them to teach and assign signals to undergraduates.

Why would this happen under a signalling model? Employers don't really need a top-tier PhD to separate the wheat from the chaff do they? It makes no sense with a signalling model.

It makes good sense if you think people go to a tough school to a large extent to get an excellent human capital investment. Signalling is obviously a part of the story, but you'd think the academic labor market would look very different if it were a big part of the story.

So what am I missing? How does the signalling model explain the fact that academic labor markets seem to sort in the same way that undergraduates do?