Several reasons. IMO …
The sensors for them are expensive and mechanically delicate. They have not been part of standard private plane equippage since forever. As a result, the vast majority of private pilots have no idea how to use one. It would simply be one more bit of panel clutter that will be ignored when the going gets tough.
By definition, a pilot who inadvertently stalls was one who at that moment wasn’t paying attention to what’s important. Typically they get mentally fixated on something else, whether that’s a mechanical failure or a maneuver they were trying to salvage more aggressively than was possible.
It’s also worthwhile to distinguish between what I’ll call “instrument-centric flight” and “visually-centric flight”.
In a jet or turboprop, you’re always flying on the instruments even if it’s a lovely day outside and you’re in the traffic pattern. Yes, you’re glancing outside for traffic and perhaps for the runway you’re trying to land on. But it’s 90% inside, 10% out the window. The value of a HUD is to move the instruments out the window so a pilot can monitor outside more while not losing the vital inside picture.
Conversely, in a single Cessna, unless you’re in the clouds you may not look at the instruments more than a couple of seconds per minute. The whole rest of the time you’re watching outside. And that can be true even if you’re on an instrument flight plan and following an instrument procedure. Ideal technique in the latter case would have more attention paid inside, but it still doesn’t need to be even half-time; things just don’t wander off the desired parameters that quickly in a slow aircraft.
The point being that for the airplanes closer to the private plane end of the spectrum, more instruments don’t lead to more safety since most of them aren’t being looked at most of the time. More skill at managing distractions and better basic airmanship so pilots can better feel and hear what’s going on would be more effective.
An interesting halfway house would be an stall warning system that was a) not just binary, and b) audible. Which would almost certainly be implemented as an outout of an AOA sensing system.
In old Cessnas the stall warning device was no kidding a 8" long child’s plastic bicycle horn that instead of a squeeze bulb was plumbed to the leading edge of the wing. As the airflow started to separate it’d softly bleat intermittently, then as the stall progressed it’d get louder, angrier, and more insistent. In a severe stall it’s be howling away. You could accurately fly very near to stall entirely by ear, just modulating the noise by pushing or pulling.
Conversely, most other aircraft simply have an electrical switch that’s moved by the airflow. Switch closed = warning light and/or noisemaker active. Switch open = nothing. The pilot gets zero feedback approaching the warning threshold and also zero feedback on how far past the warning threshold they are once they’ve passed it.
A system with non-binary outputs like the Cessna system, but that starts audibly signalling much earlier and has a greater dynamic range from “getting close-ish” to “waay stupid stalled!!!” would be useful. Once everyone was trained to use it.
The other issue with stalls is that airplanes don't stall when they get too slow. Despite that being the standard FAA-approved private pilot pedagogy. They actually stall when you try to pull more G's (including even less than 1.0G) than the wing can deliver at the current speed.
The consequence of this is that most stall accidents don’t directly result from being slow. They result from being slow-ish, then adding more back-pressure = more G loading, than the wing can handle. Or from a wind gust doing the same thing. Then once G-demanded is excessive to G-available, in just a second at most the airplane transitions from well-above stall to well-past stall. The gyration begins and the ground arrives before there’s time to do anything about fixing that. Oops.
The way to win that game is not to play. A deep-seated awareness of remaining G-available is what’s needed. You’re 100% right that an AOA provides something much closer to that exact info than an airspeed indicator does. But only if you’re looking at it or listening to it. AFAIK nobody makes an audio AOA device.
My current ride has an AOA gauge on the PFD and on the HUD. And I grew up flying AOA in USAF. In fighters you live and die on AOA. With that background …
It’s easy to see the gauge, but it’s even easier to ignore. We spend our entire career trying to stay well away from critical AOAs. So it’s almost never delivering info we can actually use at the moment. On the rare occasions I fly post-COVID I’m actually actively trying to teach my self to pay more attention to it. It’s an uphill struggle.
