Nexon: Can and Can't.

As far as compatibility goes, if NexonKR does do all of the radical changes, then I don't quite understand why the breakages occur at all.

You need to consider how far apart the versions were until recently.

If GMS requested a change to their version, a unique addition for example, it was being inserted into a version of the codebase that KMS hadn't used for months, if not years. KMS didn't then go through and port it forward into their own fork, they just left the branches distinct so that GMS was no longer evolving into KMS, but was evolving into an alternate reality KMS with it's own historical differences. When they then wanted to do an upgrade they'd graft over the code from the KMS fork from the appropriate point directly into GMS's branch. Generally when you try to upgrade branch B version 66 to version 67 you look at the differences between branch A 66 and 67, and then apply those. For something this complicated you actually have to look at the differences between Branch A 66 and Branch B 66, compare those, then compare the differences between 66 and 67 and try to merge them carefully, but the more complicated the differences between the two 66s the more horrifying the potential to miss something and create the disasters we've had. Now throw in that they were also cross pollenating from MSEA and JMS and you've reached six orders of magnitudes of complexity, because each of those is also at it's own distinct evolutionary point diverging from the base model.

For an easy example; Think of Fiel's conundrum trying to show you changes between GMS and GMST. He has to show what changed in GMS version X going into GMST version Y, but then when it moves to prod he has to pick between showing what changed from GMS X to Y, or GMST Y to GMS Y. Now multiply by six, find all the differences between every possible combination of versions and try putting them together. It's doable, but it's not quick, not pretty, and leaves a lot of room for error and oversight if you don't know 100% what everything you're touching in every version may interact with.

I believe the horrible unsustainability of that scenario is why everyone is so similar now.

I don't know the intricacies, but is it still a licensing situation if NexonKR is the parent company?

A subsidiary generally follows the same licensing model as other franchise participants, they're just a franchise that's owned by the corporation.
Some times this means the licensing is done via "funny money" (no real charge because they're all using the same pool) but in international cases more often than not the subsidiary is treated as an independent entity and responsible for billing, upkeep, etc, just as if they were another company because they have their own income stream, their own taxes, their own accounts and for legal purposes are independent.

Since they're not publicly traded it's really hard to say how they're doing it, but the more aggressive split seems likely from how they've described interactions.