Some of it depends on the age of the airplane - I’ve flown an airplane built in 1942, and many that rolled off the assembly line between 1950 and 1975 when modern automation was not a thing. As noted, the FAA imposes a LOT of barriers and hoops to jump through to significantly modify an airplane from its original equipment. It’s just not cost-effective to make the changes.
That said - some kitplanes, ultralights, and newer designs are closer to “turn-key”. The Ikarus I flew was fuel injected with digital fuel control which translates to “a lot less monkeying around with carb controls and other knobs in general”. There was still a start-up procedure, though, and it wasn’t simply turn-the-key-and-go.
This is because a bunch of that “turn knob, wait, flip switch, wait” business is to double-check that things are actually working prior to taking off. I’ve caught electrical problems, fuel flow problems, instrumentation problems, and so on prior to launch by proper pre-flight and start-up procedures. (I have also missed a couple, finding them after I was in the air. Such events can be… educational 
). Since it’s much harder to “pull over” in an airplane as compared to stopping your car if there’s an issue there is definite motivation to find problems sooner rather than later.
That said, in a dire emergency you CAN jump into, say, a small Cessna, ignore the formal pre-flight checks, and get going with about as much to-do as a car… but the risks go up considerably. I’ve known people to do that, and they even get away with it… sometimes. Sometimes it does not end well.
Now, could we design an automated airplane that could do all those little checks and what not so the would-be pilot could hop in the seat, turn the key, wait three minutes for the machine to self-diagnose, and then go fly? Yes, I think we could. But pilots are trained to pay a lot more attention to their vehicle than are drivers of cars, and by the end of that training tend to want to know more than the average car driver.
It’s a bit - very little bit - like the distinction between drivers who drive automatic transmissions with “idiot lights” on the dash who don’t give a damn about what’s going on under the hood, they just want something to take them from point A to point B reliably and safely vs. drivers who want manual transmissions and all sorts of RPM and temperature gauges on the dashboard because in addition to simple transportation they want to understand what is going on in the machine and be more involved in it.