I am not going to keep talking in circles with you about this. Try to take it all in.
1) It's a feels bad mechanic.
2) it strips THAT player of THEIR agency with a, kick them while they are down chance that they just flat out die after already surviving that chance.
3) If they have spent points on talents like lucky it kicks them twice.
4) Lucky 3 is the problem. Either don't let players buy lucky 3 or change it so that it doesn't do what it currently does.
It's not a good situation that has anyone walking away from the table with a fun story about what happened. "I got broken and then while I was laying on the ground the DM made a roll and I guess they killed me anyway. Couldn't do anything about it, I guess."
I'm afraid I'll have to disagree with your points.
The player can still crawl, their teammates can still try to heal them or kill the enemy, or do any number of things to the enemy.
The npc or monster can still kill the PC outright even if they have Lucky 2, Lucky 1, or no Lucky at all. What the player is buying is a chance to have a less serious injury.