That's the amazing thing about Rod Stewart. He almost literally had to reinvent himself when he lost his voice through cancer surgery. And - as the documentary pointed out - his "new voice" turned out to have a greater vocal range than his old one, allowing him to do covers of those old songs.
One criticism I'd have of the programme (tho' I must admit I missed some of it at the beginning) is that it tended to gloss over the bad bits of his career, but what we did get was still excellent.