HERO System 6E does, but that's a system that puts a lot of work onto players. The skills in it are supposed to be used in heroic play, rather than superheroic and it shows. The basic system is 3d6 roll under with 11- being the 'average' roll. You can spend 1 point on a skill to make it 8-, 2 points total to make it 11- and 3 points total the skill roll your stat roll. (Each skill is keyed to a stat and stat roll is 11- at 10 in a stat and it goes up by 1 for every 5 points in a stat).
After that you can buy skill levels for 2 points per +1 or penalty skill levels for 1 point per +1, but penalty skill levels only mitigate penalties. There's an 'Everyman skill' option that's just skills that get the 8- rank for free.
The Skills book just expands for it a lot and has a ton of examples and modifiers for the skills. For example, it's -8 to Charm when trying to seduce a completely biologically incompatible alien.
It also has four pages devoted to mutual intelligibility of pretty much every real life language that has at least a million speakers.
The Bases book expands on it by letting you purchase facilities in your base that help with Skills. Like a state-of-the-art Surgery room for Medicine rolls or a specialized library for specific Knowlegde skills.
You usually want to stack modifiers and the easiest way is to use extra time. If an action takes one turn (12 seconds), you can take an hour to get a +4 or +5. It's simple to adjudicate and easy to calculate, since you never work with big numbers. For things like picking locks, I personally allow adding the bonus after the roll as in "You're two short, so do you want to spend 30 minutes picking the lock or do you try something else?"
High-power superheroic characters are built with so many points that they often get 18- roll on skills with just their base stats, so the system kind of falls by the wayside at that power level. Superman doesn't defuse a bomb, he takes it and flies so high that it harms no one when it detonates.
I was about to chime in about Hero 6E but you beat me too it and wrote pretty extensively. So I'll just add my personal experience which is that in practice it plays pretty darn well. Certainly a Hell of a lot better than the swingy d20 systems. 3d6 presents a very nice curve that makes a +/- modifier more significant than its d20 equivalent. A +1 in a d20 system makes a difference 1 in 20 times. In a 3d6 system where a target is 11, it's more valuable. And a +2 or +3 in a d20 system isn't really that much more significant than a +1 but in Hero it becomes much more potent. One thing I feel isn't clear from your very good overview (not your fault) is that the book is very supportive in how to use all this stuff. Like it has a friendly little table that tells you what the odds are at different targets to help a GM.
Hero 6E is a complex game, BUT, I feel most of the complexity is front-loaded. Working out all the powers, building characters, etc. is definitely non-trivial and maths-heavy. It's an old school GURPS-era system. But in actual play, I felt it flowed pretty quickly. In fact, at least as fast as D&D once you understand it because D&D is littered with exceptions and special cases, whereas Hero 6E is more complex at base but fairly consistent.
It's my favourite system for complex, non-casual games.
I don't know, fixed initiative always struck me as a way rules retards could rig things. Good luck doing that when your minimax plan relied on acting in a certain order, something that would never happen in "reality" even in a fantasy scenario.
A couple of more interesting initiative systems I've come across in my time are Free League's Aliens, Free League's The One Ring and Cubicle 7's Doctor Who.
In Aliens, you just use playing cards, shuffle ten of a single suit together and everybody draws cards and goes in that order for the rest of the combat. With a couple of exceptions! Firstly, two friendly characters can swap cards so long as it's before either of their actions. Secondly, a good combat result can let you swap with an opponent. Which represents you gaining the advantage - next round, it's
you who'll be getting to act on 7. The system is fast, simple and classic Free League, accomplishing what they want which is fast-paced play.
The One Ring is even simpler: Heroes go first. And amongst themselves they go in order of the stances, Forward stance first, then Open, then Defensive then Rearward (iirc). Again, it's classic Free League in that it just goes straight for what they want to achieve - the PCs are the heroes who drive the action and the boldest ones first.
As to Cubicle 7's Doctor Who, it again is worth mentioning for how very well it achieves its ends. It's based on the type of action and goes: Talkers, Movers, Doers, Fighters. If your action is to say something, persuade someone, whatever, you go first (ties solved by attribute). Then movers, so if you want to run away, jump through a portal, get in an airlock, you're next. Doers come third (iirc) so you want to activate that forcefield, start the TARDIS, drink the nanomachine solution, its your turn now. Finally, if what you want to do is a fighting action, you get to go last.
It's simple and makes the game play exactly how a Doctor Who episode would. Artificial of course, but knowingly so.
Jihad on X anything cards. Oh, you're triggered? Well, we'd best protect you from yourself. Get the fuck OUT of here NOW.
I happy with a middle-ground on this. If a player has a long list or the thing they object to is a fairly core thing like "no gore" in a CoC game, then that's just incompatible. But I will try to accomodate particular triggers. Most of us have some and if you don't you're probably young. Like there's a lot of people who enjoy the Aliens movies and are fine with all the various gore in general, but still have a deep dislike of the maternity ward scene in AvP: Requiem. I pick that example because I am one. You can include nearly any sort of gruesomeness you like but hurting babies is a major turn off that will disengage me from your game
fast. If someone privately says to me that a particular theme is distressing to them, I'll try to accomodate them. End of the day, it's about having a fun game together, within reason.