Painting Over Old Walls
I feel your pain.  In some rooms I had 2 layers of wall paper and the upper layer had been painted - SWEET.. yeah! 

I have had your issues tried the sanding methods.. I tried joint compound to fix the imperfections, and used flat paint - that kind of works - but lots of work and not perfect. 

For how much wall are you talking about I would pull the old drywall and replace with new - Faster, Cheaper, Better. 

Assuming you are in the USA The drywall will set you back $10 a piece, and the joint compound about $8. Joint tape about $6 . Given it is a stair way there probably is not much dry wall .. 

After my experience here is my opinion: I would Remove and Replace (R Square for short) the dry wall with new as I think it is faster, cheaper and will turn out better and you should be able to use any sheen paint you like.

EDIT

To answer the question about the liner..

Basically a tape product - personally I wouldn't use it. 

You can however sand and then use a thick primer to cover up small imperfections.
Just don't expect thick primer to go on like paint - it doesn't, it is work - but for stairwell maybe you take the plunge - worth a shot anyway but will cost you some green for that primer.

To help you understand the process a bit more I attach a video for a drywall surface repair. As a preface to the video so you can understand the purpose and reasoning behind how it is done. In the video the reason for the thick primer is to keep the fibers down and have a surface suitable for the joint compound. It works but can be tedious depending on how much area has to be fixed. The sanded areas of your wall - well you get the idea from this video that it works similar except you are not adding joint compound over those areas.

This video will give you an idea of some fixes 
Dry Wall Paper Repair Process using thick Primer