Would be interesting to hear what the true culprit was after all troubleshooting that has been going on here.
It's right there on the release notes.
It didn't had anything to do with the GSX code, which has always been perfectly fine, it didn't had anything to do with the Couatl engine either. Since lots of affected users confirmed it was fixed by rolling back to 3.5.6 fixed the problem, we have been assuming it was something in the code, the auto-restart, the way the Navdata is received, so we added several small changes trying to make GSX as inactive and lightweight as possible on final and on touchdown, but those didn't do much, because the GSX code never had a problem to begin with!
The problem was the OTHER thing added in 3.5.7, which is the Livery Manager.
It was handling the GSX replacement Pushback for the Asobo model, when it shouldn't have. That vehicle is a bit of a special case, since it has two slightly different variants in the same folder, one user by GSX proper, and one used only as a replacement for the default Asobo tug. The Livery Manager wasn't aware if this special case and it was treating it like any other multi-livery object, so when it was running (it's always started on every update, to resync all operators), it Disabled the Asobo-replacement variant, which is required by the Living World and the AirportServices, in all cases when you don't use GSX.
Apparently, MSFS tries to populate the airport with default ground vehicles when touching down (GSX waits until you slowdown to do that with *its* own vehicles, either when you pre-select a gate in flight or the airport has static VGDS), and the failure to find the object which was disabled by the Livery Manager apparently cause the lag. I have no idea why I was never able to replicate it, maybe it requires a certain amount of installed packages to appear.
Now we separated the "real" GSX pushback from the "Asobo-replacement" variants in two separate folders, and the Livery Manager now knows it shouldn't touch the Asobo variant anymore.
The most englighting post were those from Driver8, who hinted that in fact, the addition of Livery Manager was also something added in 3.5.7, and other reports saying the default Pushback (replaced by GSX, but still a default) disappeared. In fact, the missing default Pushback was the real bug, the touchdown lag was just a side-effect of it.