The specific service is called: GoToAssist Remote Support Customer
There is also a built-in feature that I completely forgot about that will do this. In your v4 Unattended devices list, you can Reset the connection:
Today, I figured out what *may* cause "suddenly dissappeared". A firewall can block communcations and the system tray icon goes away! So, obviously, that wouldn't be a matter of the service or program being in a "not running" state - which is what I am pursuing here.
I surely can't guarantee (or even imagine) that this was the cause when so many computers were behind the SAME physical firewall and yet the results were mostly good and only one or a few were "bad".