Try ordering your network Services such that as each fails, the one above takes over, if it has a network.
In the Network System Preferences, click the Gear menu and select Set Service Order.
"as each service fails" --- but the iMac is too stupid I guess to know the Router has no internet connection...
I guess I can order USB-iPhone first, then WiFi, then Ethernet, but the WiFi to the failed-router looks good.
What a mess. Can't Mavericks check to see if a REAL internet connection really exists for each service?