In our area, some people use it, some don't. Our installer the Zapzit crystals that kill weeds long term. We also dug down the deep end far enough into the clay to get below the root line of most plants (though not mesquite trees- not that weed fabric would bother them anyways). He also put a layer of plastic under the sand to create a barrier to material mixing, but it was heavy painters plastic, not fabric, and he could not put it on the slope from the shallow to deep end as the sand would not stay on the plastic slope. The he put tar paper - the stuff that is used on roofs under shingles. Then the sand went on top of that. The tar paper and plastic are not just for weeds, but also to prevent burrowing critters (moles, gophers, etc) from coming up under the liner. See pictures below.
I have a friend who has had an AGP for many years. When they did their liner replacement last year, they used the weed fabric. Their liner was 8 years old, and for the last couple years of its life they had been patching weed holes. I guess we will see what happens as time goes on. I use the thick high quality weed fabric in my garden, and have learned that some weeds can grow in anything.
I thought about putting plastic or fabric under the rocks that lay around the perimeter of the pool, but from experience know that bermuda grass will still grow in it, so that area will just get treated with round up several times a year.