Today, however an update starts, looks like it is updating the Live Update tool itself, it gets to step 6 of 38 and then terminates.
As explained so many times, that's normal, nothing changed, and it always was like this. Each step means you have an FSDT product installed, so it will be updated. So, unless you have ALL FSDT products installed (for MSFS and FSX), you'll never see all 38 steps completed, because it won't try to update something you don't even have.
And, after the last step (of the installed steps), it quits automatically. That's how the Live Update always worked.
I can't get past this incessant update.
I don't know what you mean by "incessant update".
It's not as if the updater starts by itself: it starts when you call it so, each time you call it, it will CHECK for updates for all products you have installed. That doesn't mean it's updating them, but it takes some time to just check.
I have used the offline installer
And in the instructions (top of THIS thread), says that SOME files are supposed to always be downloaded, and it explains why. Again, nothing changed.
but I want to be able to configure GSX for a new add on airport
That's because you are starting the wrong icon. If you start from the "FSDT Live Update" icon, it will JUST update. If you want to configure something, you must start it with the "FSDT INSTALLER" icon instead.