If you've ever fallen on stairs, both carpeted and not - you would want carpet on them. I hate carpet and have wood everywhere but carpet on the stairs. When I was building my house I did a lot of research and discovered the bulk of it recommends carpeted stairs for safety. I considered wooden stairs with a runner, but I've had that before and found all the nooks and crannies were horrible to clean. And I've had a 1920's era wooded staircase (two sets actually - one for the family and guests and the second hidden one for servants that sadly didn't come with the house when we purchased it) - had to add these rubber stair treads on them so they weren't so slippery for my older relatives which sort of ruined the beauty of the wood. rubber non-slip stair treads

We replaced old ugly red ones with these from overstock. It is a sisal look. I was worried about cats ruining that faster (same texture as their scratch things). These were very inexpensive because I don't trust the cats at all but they have been leaving them alone. They aren't very comfortable bare foot but I like the look enough for the cost. They are nailed in with many tiny nails. This pic really pointed out a paint job is in order! They are shallow steps so they get kicked.

Sigh....carpets hide HORRIBLE surfaces. And you've just discovered another place where this is true - the stairs. *In THEORY (ahem....too much work for most professionals - they would skip over to a surface and be done with it), you can TRIM OFF the slight overlap nose on every single tread to create a perfect 90 deg angle for each riser/tread combo. Then you need to STRIP AND SAND down the white material SITTING on the wood. Please check for lead based paint if the house was built before 1978. (If there is LEAD....move on to full removal/replacement of the tread/risers and leave well enough alone.) Once the wood is PERFECTLY CLEAN (and that means free of the chemical STRIPPER you used to remove the white ?paint?), you can then go ahead and putty all the little staple holes and cracks, etc. Then you sand again. This thing should be as smooth as an Ice Rink! Now you are ready to glue down the vinyl and the nosing. Shouldn't take more than a week. Or you source IDENTICAL thickness wood (pressure treated for this purpose) and remove and replace each and every riser and tread. Be aware....as soon as you TOUCH these things, you need to bring the whole thing up to code. Move ONE piece and you need the whole thing to code. And code means "Finishing Carpenter" level of mastery....and legal everything that goes with the title. If you do this, you might as well have them deal with the banister, etc and have everything moved up to code at the same time - no sense in waiting. If you simply resurface the treads with new carpet, you get away from all of this. If you can resurface using the original material, you should be able to keep everything else as-is. As soon as you remove stuff, and "rebuild" then you are getting into so serious code issues. Personally, I would throw down a stunning carpet that compliments the LVT and leave well enough alone. Make it look handsome by using a runner and then fixing/painting out the treads/risers in a pretty way....but it will involve a huge amount of work. Good luck. Check the paint for lead before you do anything else.

Sounds like a plan.