Consistency of materials throughout a house makes it feel cohesive and planned. Why not do site finished red oak stained to match downstairs? It will cost more but last a century.

Laminate flooring

Vinyl Flooring

The very first issue? Your stair is likely just sub floor, hence the edge to edge carpet up those stairs

When you get to the very top of the stair? and carpet becomes a hardwood or an lVP? You have to watch for a trip hazard, due to a change in thickness. STAIR codes are very strict for a reason: :)

You would need some serious work to hardwood the stair, but may be worth a look at the costs

I agree the best would be a repeat of oak downstairs....finish tone up to you - it won't be seen from the first floor IF the stair remains some sort of smashing carpet .

You may need an LVP... for less thickness, failing you address the stair..

First IMO stairs should either match the lower level flooring or be carpeted to make a break between 2 types of floor. As Jan mentioned lots of things have to be considered whan changing stairs . I probably would do the same red oak upstairs and either spend big $$$$ to have the stairs done or just choose a nice simple hardwearing carpet for those stairs .

I’ve carpeted stairs when changing upstairs floor to hardwood to match downstairs. I splurged on really beautiful neutral carpet for stairs and still spent a fraction of what cladding stairs in hardwood would have cost. It really helped sell the home when the time came.

Food for thought . . . carpet v.s. solid surface flooring upstairs will have an impact on sound transmission. It's certainly true in my house.

this ?

or this?