Maybe!
Doug
Well that sounds like a great reason to look at Pathfinder 2e
Does a crit on an attack automatically kill the attacked thing? Of course not, that would be absurd. Depending on the rules you're using it either does max damage, or bonus damage. It also often is a successful hit even if the attacker would not successfully hit. It is the best outcome the attacker could hope to accomplish.
A crit success at a skill check is no different. You can not expect to convince the Dwarven kingdom that you, a human, are the long lost prince with a deception check any more than you can expect a first level rogue to sneak attack any noticable damage onto the Tarrasque. But you can score a hit, or convince them you believe you are the long lost prince and that maybe they need to find out why.
It sounds like what you think a crit anything is is pretty dumb. Success doesn't begin and end at accomplishing the entirety of your goal with a thing. If it did we're still going to have to make every combat crit a kill shot.
Not at all. A crit is never doing the impossible, it's doing the best possible. A crit at first level isn't going to one shot an elder dragon, but you'll hit it and do some damage.
A crit trying to lift the castle's giant, wrought iron portcullis isn't going to lift it, but it just might help you realize one of the bars isn't as firmly connected as it ought to be...
In other words of what others have already said, a crit skill check isn't making the impossible possible, it's the best possible outcome you could hope for. Just like how a crit on a thing you can't hit is the best you could hope for. You don't instantly kill it, you just get a very good shot in.
You don't convince the guard to let you go free, but maybe you manage to get him to believe you're inept enough that he can go to the other room and have a nap.
There's no crits on skill checks in the book. Play how you want
Manslaughter is still adjacent if not under the same umbrella. It's manslaughter sure, but drawing that line isn't always as neat as we'd hope.
Justice and vengeance don't go hand in hand, they have crossover in a venn diagram. What if no one in the family is willing to commit a heinous act upon another human being, even if that was the person who visited it upon their loved one? Are they pardoned? Does someone else do it?
All that is to say nothing of the effects such a job would have on someone already in a vulnerable mental state. Vengeance rarely brings relief, but other things can weigh heavily on a person.
This is one of those things that sounds good when you're in a certain mental space, but I think will largely fail under much consideration.
Lets some off to easy and others too hard
If you kill a dozen people by one shot to the head it's done quick for you but those families will carry that weight a very long time, for one example.
On the other end if your breaks fail and your out of control car hits someone, do you deserve to die?
Also the question of who do you get to carry out these sentences? Surely the people willing to carry them out should be cause of suspicion...
I think the answer you're looking for is
Yes
Become popular? It's been popular roughly for the lifespan of the format. It's hardly language's fault the developer wanted to make an unfunny reference to a since forgotten peanut butter slogan.
On the other hand linguistics indicate a hard g sound with the construction of the word, constituent words aside. Plenty of four letter words starting with the gi combo have a hard g, including but not limited to gift which you may notice is very similarly constructed.
Whatever else the English language may throw at us, people appreciate consistency because we can make some sense of the world. A hard g is the consistent, predictable, sensible choice for the limited availability of those virtues English offers.