You know, I'm surprised any bus companies bother to operate services at all, going on some of the responses you get on this forum...
Limited Stop services, at least where they are just running down normal roads with lots of bus stops, aren't popular with all operators as they often generate complaints and become a constant argument with passengers who see a bus going where they want to travel but not stopping where they are but stopping at a point just too far to reasonably walk to but too close to justify catching the other bus. It is a slightly different calculation when you aren't the local operator or there is a very good frequency stopping service in parallel (so the TP in Derby is limited stop but that isn't High Peaks core area and there is a 10-min frequency TrentBarton service to cover - High Peak moved the all stops section much further south after they took over the TP due to customer requests as the balance changes as you get towards Belper).
Unless you have a very high demand corridor it is much more likely to make a service viable by making it accessible to more people by stopping as close to them as is operationally beneficial than missing them out to save a minute or two running down the same road you would anyway - different to using a by-pass or dual-carriageway where the travel time savings may well more than justify skipping sections of route (so your suggestion of using the Airport Link Road to cut the corner is a different question to running limited stop up the A6 out of Buxton, it is just I don't believe there is sufficient volume heading to the airport from outside Greater Manchester on the A6 corridor alone to support a good enough service to be attractive as a standalone direct service even if attached as a through service off the Transpeak from Bakewell & Matlock).
My point was whether Transpeak beyond Buxton would be viable these days run as an Airport service. You'd probably not add much to the journey time to Manchester and would provide a better link than there is currently.
Airport services are complicated propositions as a standalone commercial operation when dealing with coming from a distance (a lot of them have some hidden subsidy from airports or councils to support them in some form particularly around getting operating days that suit airport workers). Most of the workers who make up the regular travellers are on shifts with a start or finish at times that are outside normal bus operation time increasing operating costs significantly to service whilst passengers are very irregular users and demand varies significantly day to day, week to week and journey to journey.
Running via the Airport isn't likely to make significant differences to the viability of running to Manchester, you aren't going to get local travellers between Airport & Manchester due to the level of the parallel train service and the frequency of the TP north of Buxton wasn't good enough to be useable for the majority of workers or flyers without a significant increase in resourcing. The TP had a handful of journeys a day to/from Manchester, basically every 3-hours, effectively using a single extra bus that fitted between school/college work taking 1hr 15 with the first bus not arriving in Manchester until mid-morning during most of the year. To produce an attractive and viable service you describe would have required 2 extra buses to allow an hourly service and 5 or 6 extra drivers to give a operating day that actually meets the needs of travellers to the airport (both workers and airline passengers) when they need it rather than just when it suits the operator.
Yes, my experience when in Stockport was it wasn't used much cross-town. It was great for me going to the airport, faster than the train as well as cheaper. I'm quite surprised there hasn't been a better Stockport to Airport bus put on post franchising, if anything I believe the 199 has been cut back on that section.
Only really a couple of late evening short trips between the Airport & Stockport that were supported by TfGM until they decided they could only provide funding to Bee Network branded services. TfGM have probably decided it only really needs a half-hourly service and if High Peak are prepared to provide this commercially there is no justification in them paying for a service on top.
However, doesn't your asserting that the majority were going to Stockport rather undermine the question of extraction by a direct airport service? You'd lose a few certainly, but the much quicker journey should also add custom.
The bus does go the other way as well. Putting a limited stop service (presumably stopping at the busiest stops) will draw passengers heading to Buxton off the all stops service which probably isn't as financially secure as you may think. Also if you abstract the low demand from south of Stockport to the Airport off the 199 that undermines the 199 running to the Airport and that may undermine the viability of the current service levels to/from Stockport (particularly the very early/late trips which are probably more reliant on airport demand due to shift workers than the core daytime service but also potentially the underlying frequency).
Being able to combine 3 or 4 flows onto a single route (into Buxton, into Stockport, Stockport to the Airport and that small demand from south of Stockport to the Airport) means you can viably provide a better service to all than could be supported by trying to run 3 separate services (if you take the small south of Stockport to Airport flow off the 199 you could destroy the viability of the 199 running through from Stockport to the Airport leaving that having to be run separately) with none of them being able to support the current service level as separate services. It does mean that one flow, the one with the lowest demand, has to sacrifice some journey time advantages but gains frequency benefits that could be seen as more important to many of the passengers.