For what it is worth, I enjoy the real time of the mini game. I'd even be happy with real time in ground combat - that is how I used to play Apocalypse. And you're right about the vocal subsets of the community who hate on seemingly any option for air combat. You can't please everyone. My reasoning is that you please the most people when the style of air combat fits in with the rest of the game. In my [inevitably biased] opinion, the best fit is something that tests similar skills to the rest of the game. So, for instance, putting in a first person perspective flight simulator would be jarring (not that I'd mind). However, I can see your point: converse to my reasoning, there is a rationale for making the mini game something that tests an entirely different skill set so as to break up the pace of the regular game. But I feel fairly confident that my logic stands, you don't want to break up the monotony with something too incongruous. On balance, I guess I just sympathise with the vocal minority who don't like to test their reactions by spamming the pause button.
I hadn't thought of the thematic relationship in this way. Sure, mapping the flurry of aerial combat onto a more chill turn-based system does have a thematic mismatch. But I think the priority of good gameplay trumps all. Look Firaxis XCOM, that had a good thematic fit for the air game, remember that? Felt exactly like you were a commander in the base watching the interception on a scanner. Great thematic immersion, but it was dreadful gameplay.
Overall, I think I just don't buy into your argument that the mini game has to be 'fast, short and easy to autoresolve'. Sure, even @Chris has said that the air combat shouldn't detract from ground combat. Still, I'm not sold on that (well, logically I'm not sold; financially, I have already bought the game). Air combat only needs to have an autoresolve if the mini game is not interesting enough justify making the player go through each and every battle. My reaction there is to make it less tedious or occur less frequently. X1 had the issue that you were swamped by air combat. Lots of instances of the mini game. That undermines it being a short and fast distraction from 'monotony'.
As OP says, air combat in this genre of games has been a mess. XCOM2 was smart/timid enough to sidestep it entirely. Looking beyond issues of mechanics and design, I can think of three points that that part of the game addresses (apart from adding diversity):
To spawn ground combat missions
To test your R&D
To force you to expand
(maybe: to demonstrate the invasion and its increasing intensity??)
The first and second points don't require air combat. There are other ways to spawn missions and the gear you can give your troopers is test enough of your R&D. XCOM2 had its own mechanic for territory expansion which is why it could make its sidestep. Indeed, for Xenonauts, it really is that third point that matters most. The bulk of the challenge of the air war is getting your interceptors there in the first place (and to a lesser extent, making sure they're well armed). As such, most of the air war is determined before the air battles even begin.
So where does that leave the air combat mini game? Well, I think the design should be tailored so that air combat is better at addressing the first two points. So, the decisions you can choose in air combat are a reflection of your R&D pathway, and the goal you're working towards is the type of ground combat mission you want to spawn. With that in mind, your suggestions about diverse equipment for interceptors and locational damage for UFOs are solid. Real time or turn based; I couldn't care less. Thematic matching; meh, would be nice but is not essential. But I reckon the core of air combat should be on an ability basis, and that for me the real immersion breaker is that your fleet of a dozen condors can rule the skies.