He can add the same flooring on top of all the existing treads and still have the same rise as he did before he started.
The treads will be solid, Ive seen it done many times over the yrs.
How do we know the stairs arent a pre-made common set of box stairs whipped up in some factory. The stringer may be routed out for the treads and risers with wedges pinned in. It could be a bigger night mare if he's a diyer novice. He would need to start from scratch. Im sure just adding the flooring on the treads and building out the riser towards the nosing will be less of a head-ache for him.