Update 27-Mar-2018
I did some more digging today and found the following:
https://issuetracker.google.com/issues/72450185
So it appears everybody's getting pushed on to the Google Cloud SDK. This link also includes instructions on migrating over; I haven't tried this myself yet, so good luck.
Partial answer:
In SDK versions 1.9.60 or 1.9.61, the method
void setJarSplittingEnabled(boolean doSplit);
was removed from com.google.appengine.tools.admin.AppAdminFactory.
When deploying a project, somewhere along the way, this method is being called and obviously doesn't resolve, the deployment aborts.
I can only speculate - looks like jar splitting isn't a thing any longer so it was removed from the SDK. As this is happening on a very old project of mine, I suspect somewhere buried in the project settings is an old switch that triggers the call to that method. Easiest (not necessarily best) way would be for Google to re-introduce this method as a stub that does nothing so we can continue deploying our projects using current SDK versions. Or of course if somebody can point to a setting somewhere (that I couldn't locate), that would be even better.