Okay, there is nothing really good about this idea at all.
Doesn't matter what you call the lowest rank, it's still lowest.

Make the system disabled by default and completely optional.
Right, optional. And if players demand you use it? You get locked out of parties for not.

Make the system personal only, no way to see any other party members ratings.
Again, you offer a solution, but you previously already stated what is going to happen. People in PFs are going to demand screenshots for "proven" skills. And if the skills aren't good enough, the player gets kicked.

• Makes content more fun to replay.
No, it adds stress if your rating is low and you start losing out on PF content because of elitist jerks.

• Encourages improvement.
By helping other players judge you and tell you that you aren't good enough for their group.

This system would give more incentive to players for re-playing old content
The incentive is already there. It's called adventurer in need bonuses, tomestone bonuses, stellaclusters, etc.

DPS roles can have their criteria based on their total damage output at the end of the duty + the effectiveness of their buffs. Damage output thresholds would be set accordingly for each duty.

Tank roles can have their criteria based mainly on damage taken + damage mitigated, whilst also taking into account their damage output and losing points for total damage taken by other party members.

Healer roles can have their criteria based on both their total healing and their damage output. The key is to find the right balance so as to reward healers for playing their roles both when they are spending most of their time healing a party that is taking lots of damage, and when they are spending more time dealing damage due to the tank effectively holding aggro and using defensive abilities.
You really haven't put as much thought into this as you should have.

Tanks: Their rating is lost because the party taken damage. Does this take into account party members stepping into AoEs and getting hit? You realize this can be a way to torch a tanks ratings by deliberately taking damage, right? So lots of opportunities for grudges.

Healers: Your system for this is again bad. In dungeons, most of the time the tank will pull large groups and sure they hold aggro, but you are backwards in this because when the tank is holding aggro for a large number of enemies, it requires almost constantly healing to keep them alive from taking many hits repeatedly.

DPS: So you want to shame them for either not understanding what rotations to use. Or else force everyone to play exactly the same with the exact same rotations of damage and buffing themselves, or the group. Or else they are seen as weak and undesirable in a group.

Sorry, but this idea, frankly, has no business being in XIV. There is just no way this won't negatively affect the community.