I'm glad you are asking this question on Houzz. I agree with your thinking that the floor vinyl is aesthetically superior. I removed the carpet from my stairs and replaced them with wood overlay, because I could not safely vacuum them to be really clean. Others think they can with a hand-held vacuum. My husband commented afterwards that he liked the wood stairs better than carpet because the carpet was too slippery. Go figure. Ask the manufacturer about the slip resistance rating (DCOF) of your choice. This might help: https://floorchemdepot.com/is-vinyl-flooring-slippery-understanding-slip-resistance/

I like stairs and floors all the same and many LVP compnaies now do stair nosing to match too. I too find carpet slippery when wearing socks

Vinyl is the worst possible choice for stairs. And it's not great for floors either. Not unless you pay tens of thousands for remediation to make the subfloor FLAT, and choose commercial level glue down. Not click. You pay a huge amount of money for something very temporary if you put it on stairs. In 3-5 years, you rip it out and install real wood. Skip the interem years and expense and go to real wood.

Can only speak from personal experiece. My brother, wearing socks, slipped on their wood floors, broke his back. I slipped on the wood stairs at my previous home, matte finished walnut, thankfully not injured, however think that a nubby textured berber, or something like that , would be nice as a runner on the stairs. Patricia’s suggestion sounds good.

I prefer wood stairs but would include a runner. Wood floors are too slippery in socks.