Back buttons would be another nice one. Right now there's just 2-4 buttons too few on controllers, and it often leads to strange button mappings that either shift with context or require multi-button activations, which gets even more annoying if you have to do it during, say, a jump.
Gyro aiming completely solves both fine aiming and tracking aim on a gamepad when paired with some kind of touch sensitive control for enabling the gyro (natural recentering).
In console FPSes they just automatically track the enemy if they're near your crosshair and call it a day-- giving everyone an aimbot instead of solving the UX issue.
For most people you're better having relatively high sensitivity on the gyro and using the stick for large movements. Using human pistol aim as a metaphor it's like the stick is your arm, and the gyro is fine tune aim in your wrist.
I've first played Zelda BotW/TotK (which is very light on precise aiming), and I found the gyro both precise and intuitive. The game is nowhere near as fast-paced as a modern shooter, and the weakpoints are large enough to consistently crit. I enjoy the bow.
Then I've Switched to Warframe - a looter-shooter. NO auto-aim. My first attempts to aim with the thumbstick were painful and felt pointless. The default sensitivity was very low, which I imagine was supposed to help aiming, but it made many parkour moves near-impossible (the game heavily relies on both). You could always press a button to place the camera behind your back, but that was two-step, non-incremental, and wouldn't help turning up/down.
So I've cranked thumbstick sensitivity to the max - turning the camera whichever way was now easy; then committed 100% to the gyro for aiming. Honestly, I'm much more precise than I've ever been with the mouse. I can consistently land headshots (super important with incarnons), use bows / thrown / charged weapons, etc. My hit ratio is between 50-70% for most weapons.
I'd now be hesitant to aim with a mouse. Thumbstick - out. But that's just personal experience.
It's extremely complicated however (like many things USB), which is probably why everything just emulates an XBox 360 controller like you said.
Especially if you supported both XBox and Windows.
So the only complex HID game controllers are for very much enthusiast setups, which are rare enough to trip things like absurd assumptions in HID drivers in some systems (the joystick+throttle I have used to break linux HID driver because someone decided to statically allocate possible amounts of joystick buttons per device...)
Hell, configuring my own controls for a game is one of my least favorite things to do. I haven't even played the game yet, I don't know what button should do what!
The way it is, the devs know what kind of controller everyone will likely be using, they can figure out their ideal mapping for how the buttons should be used, and we all have an easy time using our controllers.
I realize this may not be practical, but it's kind of weird that PCs have been more or less stuck with a protocol designed for XBox 360 controllers for 2 decades now, while the locked-down console space is seeing much more experimentation and innovation around input. The original steam controller at least hinted at being sort of an open platform for this sort of thing, although it didn't really take off. Fingers crossed for the new version.