Correct. In the case that it stops transacting, the value of bitcoin as a system would be zero.

In the case where the costs are larger than the value produced, but are not paid by the bitcoin system (external, you might say), then it would still have net negative value, but continue to operate.

The value wouldn't be zero, the transaction rate would head to zero. The value would be stuck at whatever it last traded at, so it'd become very volatile.