That would have been my concern. Leads me to think they may not have been at critical temp. It's also possible that they added the chain once the anvil was up to temp. At 1500+ deg. F. steel is very very plastic. It's hard to tell from the vid, and colors never do come through right, but I suspect they were not quite to critical temp with the anvil. Also didn't see any air blower to feed the fire, but that might have been off to the side of the vid.

I believe I'd have made a chain lasso and looped it around the waist right before pulling the anvil from the fire and quenching. In any event, using the river and an excavator to quench with is a very good idea.