You cannot cover them with wood treads. Wood treads would replace those. They are to thick and will make your stairs unsafe as every step needs to be the same rise and run and adding the wood tread with make then rise illegal (and conversely unsafe).

@millworkman what would be your reccomendation? Peeling off the laminate and just refinishing the existing treads?

You may want to look in to the possibility of using vintage runners if you like boho style.

You’ll never regret completing a project the right way…you might regret the extra time and money you hadn’t planned on spending, but isn’t it better to leave a space better off than you found it. I’d remove the linoleum (following whatever safety precautions you think) and once that is gone you move on to the next step, whatever that may be when you see what’s underneath.