No, they would have the same ability.
The reason they locked him away was not because he had any special powers, but that he had charisma.
Charisma is a far more deadly weapon than any ability - to be able to convince others and get them to do your deeds is a potent weapon, especially when you are practically invincible.
Immortality, in reality, is spontaneous regeneration. The way he heals is exactly the same as Claire, so stop splitting hairs. Also, because of his ability his cells would be immune to random mutations, ruling out him having any other abilities. Same goes for Claire, as shown by the massive radiation exposure.
I would say they need to find Claire because when you take blood samples and DNA, you don't tend to take the anti-bodies as well, just a blood sample and perhaps a cheek swab (after all, he was one of them). They didn't know they needed a cure until they had tested it, as they assumed they had a cure.
No plot hole, just stating the obvious facts really. Now stop assuming that everyone needs an ability to do everything, it's really bad form to ignore the basic history of humanity in favour of powers.
And 400 years is a long time to practice his charisma skills