The issue is most people think that if its frozen you can just kill it with weapons and it counts but that's not how it works. Any weapon even kinetic counts as weapon damage and not stasis. You have to freeze or slow the enemy, weaken and then kill with a punch, grenade or super. Usually most people going into nightfalls are doing stuff like this. If I see someone using stasis on one I usually help them out. A lot of people however are done fragments and don't care.

Solo legend NF as you say. Take the right element to get you to the first champion, stasis finisher it. Rinse and repeat. You'll get it in under 45 minutes.

Spam the finisher button. The default works as does the finisher awarded for hitting Rank 10 on the season pass.

Honestly I was confused at first too. As soon as I realized that finishers with a stasis subclass count as stasis kills, it was smooth sailing. Just bring the right mods like overload rounds or unstoppable rounds or anti-barrier rounds, and when the champion is low enough on health, press the right stick to activate your finisher. It helps if you get close before their health is low enough since there is less chance that they will die before you reach them. Tbh the worst part of champions is that freezing them will cancel your interrupts. If you fire an unstoppable round and stagger a champion and then someone freezes it, they won't be staggered anymore and you have to fire another unstoppable round at them. It's especially irritating when someone activates their super and keeps freezing the champion over and over, doing like no damage and wasting everyone's time instead of just letting you stagger and kill it.

Are you on PS4? If so, set your date back, choose the heroic level and get it done quite easily. My wife and I did this today in the Warrior Hunt.