UPDATE:
I have a program installed on my computer called "Boom 3D" (v 1.1) which is an audio output enhancer. Essentially it creates a new audio output option called "Boom Audio" when the application is running (Img 1).
Oddly though, this output option is NOT available under the volume menubar item list (Img 2).
When I quit the Boom 3D application, select my desired headphones (speakers) output, then restarted Safari, audio playback worked on all websites, even when I restarted the Boom 3D application. I'm sure however that if I were to restart my computer, the system audio output selection would get screwed again.
Later I will test this out on my laptop as well (also running High Sierra, Safari 11, & Boom 3D v1.1) & will repost the results.
I doubt that the issue has to do with the Boom 3D application as I doubt that we all are coincidentally using the same application. But, this does point to a system or safari software bug that causes Safari to not recognize any preselected audio output -- thus not knowing where to send audio -- until the desired output is reselected from the Sound System Prefs pane. Again, I'm going to keep playing with it on my systems to see if the issue and "fix" are reproducible.