Subler worked. What it appears is happening is that the new version of Quicktime will not play certain encoded videos that have "hints" embeded in their file. Downloaded videos often embed these hints to somehow ensure they download correctly.
Subler is a free application. The link is in a previous post. It takes a few tries to understand it. Basically, you open the video you are trying to fix, and it will show the video, the audio - and in my case - 2 additional unnamed "hints". I deleted those hints through Subler, and they were remuxed (?) a process that took less than 1 minute (as opposed to much longer to convert the entire video).
The end result was it solved the issue.
Apple would be advised to somehow explain this -- because I know many other Mac users who have been scratching their collective heads since the Mavericks update. You simply cannot believe how you depend on Quick Look - until it doesn't work with your files. I do a lot of work with video - and this was maddening.
Problem solved. All good. A big thanks to Block Rocker