Personally, I think it’s criminal to cover up these aged and worn steps. Looks like you do too. Have you chosen a colour for the carpet yet or is that also not part of the deal? 😉
I’m obviously not privy to how your relationship functions and the subtle nuances and currencies when negotiating everyday stuff, like interior decor and soft furnishings. I do hope you’ve bargained well and made a few gains along the way? 😉
On a more serious and thread title note, I feel that anything other than an epoxy cement will not bond particularly well to the shallow depressions. Anything that does will make it more tricky to restore once it’s been decided and ‘agreed’ that carpeting them wasn’t such a cool thing to do, despite the H&S risk assessment. Ply and batten possibly chemically fixed, rather than mechanically.