Good to know! Yeah, I usually go either to that big stump near Hyrule Field or do a Lynel hunt down to the Eldin Great Skeleton to replace my Great Flameblade when it breaks.
That's pretty lucky. I had one time that a Silver Lynel dropped 2 of them.
Is it consistent that way? I've had issues trying that in some areas in the past. Besides, the other way saves on wood and fire kindling materials. ;)
On the subject of finding star fragments, I found what seems to be a nearly 100% consistent method of finding one per night. I can't take full credit for the find since I read about the basic concept elsewhere, but that method involved every full moon whereas mine seems to work every night...