General Chat

Yes. Pretty much that. That is a very good explanation.
I have looked at Neon City Overdrive, but not finished reading it, nor found the motivation to bring it to the table. It looks 'good, but nothing special'.

I have not looked at Star Scoundrels. It looks interesting, but I will wait till it is done, I just don't have the time/energy for beta-testing. I like the 'focus on relationships' part, but have been burned by Flying Circus, which promised putting relationships first (and they are important especially to the leveling mechanic, which is cool and requires one to take Stress) but ended up being pages and pages of airplane combat mechanics. :(
I glanced at it. In an effort to 'simplify' things, it removes a few components that are quite important to BitD. Position and Effect are the ones I recall at this time. While they are a bit tricky to get one's head around at first they are a big part of the system and work really well in the context.

BitD is a 'Swiss-Watch' type game, it is not easy to tinker with, and small changes break many things down the line. I suspect this is why so many of the FitD games out there are less than stellar. The only ones that I even think of bringing to the table are Scum and Villainy and Band of Blades (both by Stras, who really understands the system), and CBR+PNK, which slightly broke things in its effort to simplify, but still works well (I have not yet read the second version).
I keep hearing about Outgunned recently, —but not in time for the kickstarter. I plan to take a look at it.
'Push your Luck' mechanics can be fun (a big part of THDAW, with the added benefit of a card stack showing us what could have been if we had pushed more:). But may not be ideal in PbP, since it can slow things down as each 'roll' requires more transactions to conclude.

Have you looked at Push? Its base mechanic is push-your-luck dice done quite well.
I have not gotten round to looking at that system.
I adapted the FFG dice to a FKR system, to add flavour to the occasional rolls.

I have not actually managed to read Genesys, I hear they addressed some of the troubles that FFG Star Wars had, but I never got that system to work for very long.
Knave is my go-to for 'OSR' stuff. I do plan to look at 2e when it is done.

I am not sure if you are asking about how MGME works, or about how I would use it in a GMed game. I am sure you could find descriptions of how MGME works, and it has a lot more than just the Fate Chart (which is the only part of the first edition I remember using, though the second edition has a lot of other cool stuff I would probably use).

MGME is designed for solo play (one player, no GM), though that also works for replacing a GM in a group game (I have never seen it used this way, though:). It has many Oracle Charts and Random Tables and stuff, but, as I said, I only used the base mechanic so far.

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The Fate Chart is the main 'does this work?', 'do we win?' resolution mechanism. You ask a question (ultimately 'yes/no', but that is often disguised behind something that seems more complex:) and decide how likely it is, then roll and compare your result against the % for that likelihood.

There is also a Chaos Rank (1-10) that goes up, over time, as your character(s) lose control of the scenes and goes down as they gain control. The Chaos Rank affects which column of the Fate Chart table one looks at, it adjusts the percentages. If the Chaos Rank does not apply, use the middle column (5).

I have found that, when using MGME to explore new game systems, I often find myself preferring its simple, singular mechanic over what the game gives. The second edition even warns that this might happen, but I don't think it is a bad thing... except when it prevents ones from seeing how the new system works, of course.

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I have thought about taking 'Character and World Creation, and Leveling' mechanics from a game where they are good, and using the MGME Fate Chart when players need to resolve things. We may or may not include a Chaos Rank mechanic, depending on if it fits the theme, and it can be ignored in situations were it does not apply.

The downside of this in PbP is that it requires some pre-roll negotiation, but the upside it that it can replace some mechanics that are very slow to conclude.

For instance, combat can be resolved in a roll or two by asking the questions:

"We outnumber them. Do they flee?" Set it as 'Unlikely' (which is only slightly worse than 50/50).

Failing that we can then reassess and ask:
"Do they beat us?" Set it at 'Very Unlikely' (since we outnumber them, but the Chaos Rank pushes things towards 'yes' answers, so the direction of the question matters.)

If the above (one or two) questions conclude the combat, we can ask about if we paid significant costs (injuries, loss of ammo, whatever is appropriate in the fiction).

If the game system has a 'damage' mechanic that we want to keep, we would adjust the odds/likelihood based on how much damage we do with each hit and how hurt the enemy already is, and so on.