Think the extra die roll (like advantage) is more in line with 5e. It’s more straight forward, and is easy to add into the existing structure of your standard 5e game. If you want to increase the probability, maybe feats, or class abilities could do that?Thanks. My suggestion was designed to avoid the extra die rolling (and I must confess I personally believe rules need to happen more often than once every 400 rolls for me to be able to justify including them), but otherwise accomplishes much the same aims.
Assassin already gets a critical in some circumstances. The champion fighter has an increased critical range etc.
This maybe more something for rulings not rules?.. but you could also allow an aimed shot (made at disadvantage) to allow for just one standard critical hit to drop a target to zero hit-points.
I think as it stands it’s a good basis to work from, and changes the expectation of the game in a simple way. Gives it that edge. But doesn’t overwhelm. Strikes me as the right balance after discussion here.