Not only are good runners expensive, I think you would have a real problem with tripping on the stairs. I have had 2 wool runners installed in 2 houses and both times the installers said that you have to be careful with the thickness of the rug pad underneath because even a fraction of an inch can throw a person off balance (we are so accustomed to a particular stair height). Just a thought...

Be sure to sell or give away that beautiful old railing and not throw it on a heap. I happen to like it. But indeed, it might look better in black. The railing you have posted instead gave me some shudders. Quite a contrary style and not one I'd welcome in my home. Way too modern for a ranch home, is what comes to my mind. If you are really going to get a new railing, I like the one Judy proposed, although I wouldn't want it in white at your place. Perhaps in black and the railing that matches the wood of the stairs? Or maybe something like this?

Most economical = carpet. After that, a solid finish (wood, cork, vinyl, tile, stone, etc) will require all the extras. Painted stairs will work...with plenty of elbow grease and with the FULL UNDERSTANDING the paint will wear/chip/"patina" inside of a few years. There are ways to lay a runner on the stairs and then trim out the edges of the tread/riser with wood-looking stuff. This will cut the cost in half of what I mention above. The other option is to place the runner and PAINT out the edges....creating a faux-painted-wood look which could be very handsome yet extremely practical. Notice how carpet keeps on coming up again and again? Yah. There's a reason for that. Those are builder grades stairs made for carpet.

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.