Right then your rules triggers are wrong:
The item doesn’t receive a command nor does it changes
Updates of items to the same value are not shown of the events log.
Change the trigger to received update
rule "mechanical door bell pressed"
when
//Item MechanicalDoorBell received command or
//Item MechanicalDoorBell changed
Item MechanicalDoorBell received update
then
var msg = "The mechanical door bell was pushed."
sendBroadcastNotification(msg)
playSound("doorbell.mp3")
logInfo("SAH", msg)
//say(msg, "voicerss:enUS", "chromecast:audiogroup:PublicSpeakers", new PercentType(100))
end