Let me explain this to you as a Lux player.
For one, as many have pointed out, her ult is dodgeable and the snare to be ulted is dodgeable.
Lux's kit's power requires that she uses ult. Without her ult, she pretty much just pokes relatively negligible damage(unless she's ahead) or you have awful positioning. So what happens if she misses her ult in a team fight? Then she waits for her ult to be up which means she's not contributing to fights much.
So what happens if you kill someone with her ult in a team fight? Then she still waits for her ult cd. The CD reduction is so that a Lux that does her job can continue to be relevant in team fights. You can go "boohoo why does she get 2-3 ults in a fight," but for her to do so, she needs to be winning that fight AND other champions have abilities at their disposal to use without waiting on a CD. ADC can just auto and crit for "900" damage consistently throughout the team fight.
Lux's ult is pretty much her main damage in a rotation, the resets are just so she can keep doing her job if she's doing well.
The assist change was made because recently, due to shield, hexdrinker, supports, that one rune that's pretty much hexdrinker, Lux's ult doesn't always execute, even on a squishy. Now with this buff she gets the reset if the ult would have killed the enemy but your team is who gets the last hit.
Some ults are designed so that it's what a champion needs to finish an opponent. I mean that's how assassins work and their ult CD can be even lower or fully reset.