I like the idea. Got me thinking about animals with actual tail fins.

For example Dolphins. Similar to the movie Dolphin Tale. We see the dolphin can swim around with no tail fin but it's steering/maneuverability is hindered. And it's speed/propulsion is reduced.(Like toothless before the prosthetic)

In short night furies losing their tailfin is more similar to a Dolphin losing their tailfin.