Gary, that was nine months ago. Have the Apple programmers resolved this problem?
There is nothing in the discussion that points to errors with the application, see the following explanation.
I have a self-playing slideshow, 7 slides, each slide itself has up to twelve builds, finely timed to build in and build out while a song plays, at the end of which the slide automatically transitions to the next slide, a new song starts, the builds come and go until the end of that song - same again and again, all the way through.
What your describing is not a Self Playing presentation. If you have set different durations for each build, that is not a Self Playing presentation (Inspector > Document > Document: Self Playing)
Self playing presentations set all builds in the presentation to the same duration.

I cannot set everything to "on click" as you suggested nine months ago, I would need a script, a producer and a stopwatch to try to come close to what the timings must be.
Using on click would be the wrong thing to do in your situation, that answer was for a different set of circumstances.
I also cannot get the "Record Slideshow" under the Play menu to work. At all. It looks like it wants to being playing, but with green bar - Wsxape gets me back to edit mode (no lock-up), but no way to "record". Still no fix?
No fix needed: from what you say, Keynote is set to self playing when trying to use record slideshow, therefore set Keynote to Normal (Inspector > Document > Document) in the Presentation Type dropdown menu select; Normal, then use record slideshow.
Both Keynote and Quicktime are well-established Apple programs, can't we expect that the programmers will talk to each other and get this done? Is there hope?
I have no difficulty in creating exactly the type of presentation as you describe, there are no issues within Keynote that prevent exporting to Keynote. From the information you have given, you are using the wrong settings to export to Quicktime:
Instead, change the the presentation Normal (Inspector > Document > Document) in the Presentation Type dropdown menu select; Normal, then export to Quicktime.

Try this and report back.