Short version: Disable QuickTime Player (see breadcrumb below).
Long version:
Seems there's long been a problem with .mp4 and other Apple-associated files in Moodle. I think I figured it out.
"Files with extension *.mov, *.mp4, *.m4a and *.mpg. Requires QuickTime player or codecs," says Site administration ▶︎ Appearance ▶︎ Media embedding > QuickTime Player (at bottom, in "Legacy Players." )
I'd earlier inspected the video embed element and noticed it was indeed calling up QuickTime when it played the embed--while the player itself looked like FlowPlayer.
Figuring that might account for the auto-play audio, even when the video embedded was not itself autoplaying, I simply disabled QuickTime Player on that same Media Embedding page (see breadcrumb above).
Reloaded, and an angelic silence met me.
But note: for all I know, disabling QuickTime might have negative effects I'm unaware of.
Let me know if it worked for you too?