I still think this goes back to a network issue, and not a Mitel issue, the Mitel becoming unresponsive is a symptom and not the underlying problem... Something is happening in the network and it's locking up the embedded switch in the controller somehow.
Look at the commonalities here... You have replaced all the Mitel components except the database, but you have done a backup and restore which would eliminate any corrupt information, so the cause is likely something common to all phones that is NOT the Mitel controller, thus the network infrastructure. If you also control the network infrastructure, start looking at switch logs and making sure your STP/RSTP settings are consistent throughout the network and there are no loops anywhere. If you don't control the network, your in a tough spot as no one likes to have the finger pointed at them...
I am not saying any of these things are your issue, but if you have eliminated everything you can it might be time to call in tech support, too bad it's 10 times harder now than it used to be since we have to go through multiple layers to get to real support (I had Mitel for that).