Another idea came to me, and this is totally grasping at straws.
I noted that you say this works with other computers in the house. I am unsure if you can utilize Internet Connection Sharing with a USB tether...but...
On the system where this is disconnecting frequently, does it have a list of connections under your Network tab?
IE, has this PC been connected to a beacon, repeater, wireless access point that is still in use in the location and does not have internet? And to further that, were any of the possibly existing prior connections checked on the "connect automatically" tab?
I noted that you say this works with other computers in the house. I am unsure if you can utilize Internet Connection Sharing with a USB tether...but...
On the system where this is disconnecting frequently, does it have a list of connections under your Network tab?
IE, has this PC been connected to a beacon, repeater, wireless access point that is still in use in the location and does not have internet? And to further that, were any of the possibly existing prior connections checked on the "connect automatically" tab?