YES, possession of a seat assignment is the best way to avoid IDB. You may think there's some sort of detailed algorithm driven by fare basis or whatnot, but when it comes right down to it it's musical chairs. For my two IDB's over the years, it was simply because I was the guy with no seat assignment. (Non-elite flying NW/DL, for what it's worth. VDB's were not sought - they simply bumped the people who didn't have seats.)

But no, seeing a completely full seatmap at booking time does not mean that the flight will need to IDB or even VDB.

In my years of flying UA and AA, it seems like those carriers at least make a modest attempt at VDB before they go to IDB.

Obviously if you fail to show by the CoC check-in time, you're simply a no-show...not a denied boarding. At that point, they will usually put you on a later flight ("flat tire rule") but owe you nothing.