Oh man, that looked promising! I didn't even think about looking in Google Home. Unfortunately when I found those toggles, they were already turned off. Thanks though!
I poked around and saw a bunch of people with this complaint. In another HA forum people were using scripts to defeat the sound by muting the device, connecting, unmuting, and then playing the desired sound. Good grief!
EDIT: Nope! Doesn't work. The first connection always makes a startup sound even if the device is muted. Subsequent connections within a short time do not make the noise -- that is why I thought the test worked.
However, I DID get this to work in Rule Machine.
First I tried setting volume to 0, playing the idle sound, and restoring volume. But, for some reason, it still blooped. Muting the speaker, then unmuting, setting volume, and playing the track worked.
I hooked it up to a button and mashed it a few times and it seems to work reliably.