After I posted my reply above, I continued with my research and found this post, Captivate 7 html5 start screen blank - fixes? I've also been experiencing that issue with Auto Play and blank screens, and the auto play button remaining over the content after the slide show starts. Sreekanth replied there with this Adobe blog Update on Captivate HTML 5 content playback issue in Google Chrome Browser. I followed the directions there and applied the index.html patch to my Cp7 and Cp8 installs. I then re-published my tests along with one of my actual lessons.
The buttons are now working as expected in my initial tests with Chrome for both Mac and Windows, across different LMSs. I'm trying be organized and test all of the browsers and platforms I have access too, but so far it's looking good.
However with Firefox, if my lesson uses a Smart Shape I'm still getting blank slides with the error "This browser does not support some of the content in the file you are trying to view." Another thread on this here.
I'm still testing and will report back here with my findings.