I can't speak for all stores, but most of them have a back "Genius" room where techs actually do repairs. Most parts aren't kept in stock, which is why the turn-around time for repairs is usually a couple of days. Things like logic boards screens and case parts are shipped in, and a genius will do the work in back of house.
On a normal shift, the genius probably isn't sitting on his tush the whole time. The same technicians that perform repairs are also the ones that conduct appointments at the genius bar, so they do a lot of different tasks.
Apple could do in-store screen replacements for things like iPhones if they wanted to. But in the same vein as your post, the sheer volume of screen repairs coming into an Apple store would require a full-time technician or would eat into the time spent on more skilled repairs.
I'm sure Apple has done an economic analysis on the whole situation. It's better for all parties if Apple just exchanges the device in store. If a user comes in for a repair, he will pay parts + labor. In all likelihood, Apple charges the equivalent for an out-of-warranty swap. Apple saves money on the labor (paying the Chinese factory), the user gets immediate turn-around, and people who need other repairs don't have to wait while the geniuses replace a hundred broken iPhone displays.
//I worked for Apple Retail a while ago.
+1, however all of the stores I've been in (4) have had large service part inventories on hand. Some parts still had to be ordered but more often than not we were able to grab a part from the back. Turn around time was more dictated by repair volume, if we only had 3 or 4 Macs to repair then we'd try to quote same or next day, 25+ Macs to repair then 3-5 days, and so on.
In regards to repairing iOS devices... They started repairing iPhones in store back in 2009, throughout their retail chain. It started with display replacements on iPhone 3G's and persists through today, where you can have cameras, vibrator motors, batteries, etc replaced as well.
The problem is that it's not always time efficient to replace the components while the customer waits. All of the iOS repairs are classified as 0-30 repairs, meaning the absolute longest they should take is 30 minutes. The majority can be done within 10-15 minutes by new technicians, 5-10 by seasoned techs. However, add in time for paperwork, appointments running late (see threads about how Apple turned away people who are 3 minutes late, etc), not enough staff to handle volume, etc and you find yourself in a situation where it's worlds easier to just replace the entire device rather than isolate and replace a component. Not to mention, over the years Apple customers have become extremely spoiled in expecting a new device each time they have a minor problem and expect to be in and out within 10 minutes.
/rant from ex-Genius
In regards to the OP. Yes, Apple is very strict about not allowing it's AASPs (Apple Authorized Service Providers) to work on iOS devices, whether they are in or out of warranty. Obviously they can't stop non AASPs aside from legal action for using bootleg parts, but they can revoke AASPs certification and cripple the business. The reasoning I always heard for this restriction was that they don't have the same level of quality control over AASPs that they do with their Genius staff, which is absolutely true. I've heard from several sources, who would be in the know, that the iPhone/iPad is the most ESD sensitive device that Apple has produced. During Genius training there is a great deal of time spent emphasizing ESD safety techniques and the potential for ESD damage in even the most minor of repairs. The same training materials are available to AASPs for technicians attaining ACMT certification, but hearing it through in person lectures is much more effective than reading it on a screen. Simply put, the AASP I worked for and the few AASPs I've visited have all been much, much, much more lax about ESD precautions. This has always stood out as the reason Apple goes after AASPs that start offering iPhone/iPad services.
source - worked as a Genius for several years and as a technician for an AASP that almost had it's certification revoked when they started doing out of warranty iPod/iPhone repair. That AASP can now sell iPhones and offer replacements but they are still barred from actual repairs.