Looks to me as though all the slates are on upside down. The nails shouldn’t be showing anywhere. This is likely to have been done by someone bodging a repair in the past. Water will always get through the nail hole.
Nails should be covered by the slate above and then the joint of that should be covered by an offset one above that. Its called triple bonding.
So that indicates the second option as the better choice, but you may have to replace some of the slates as they could have holes in inappropriate places.
Its not difficult to sort but you may uncover more rotten battens as you have been advised.
Go up and look when they strip back to see the real extent of the damage.
Good luck! (why not turn this into a footflaps saga?)