There are two parts as to why this happens - & both the existing answers are in the ball-park as to what these are - but I wanted to add this with addition of a partial 'cure'.
It's true that wax from a car wash will build on the glass & be partially cleaned off by using the wipers. It's also true that the wipers over time will actually microscopically 'damage' the surface. This has nothing to do with glass being 'liquid' or the relative hardnesses of rubber vs glass; it's entirely due to repeated friction. Whenever your wipers get worn by this friction you replace them - the glass, however, has to suffer years of fresh wipers.
Consider how hard the steel is on a cut-throat razor & what a barber uses to keep it honed… a leather strop. Soft polishes hard, given enough repetition.
To make this worse, you are usually dragging small dirt & grit particles across the screen with every wipe, too.
OK, down to the actual issue rather than the causes of it.
Your unworn area is waxy, causing beading, but it's not beading very well, it's still slightly 'sticky'. Your worn area, however, is imperfectly clean & still has some kind of residue on it. This is causing really bad streaking. That's the dangerous bit.
Streaking on every wipe means you almost never get to see out of properly clear glass in the rain. If you were to squirt the windows on a dry day, then let the wipers pass over a few times, your vision would be impaired until the wipers stopped & the glass had chance to dry up. This is not a great thing when driving in the rain.
The fix - & yes, this is a product recommendation, though feel free to try another brand, if you can find one.
It's a water repellant, so at first sight it looks a bit like the effect you get if you get wax on your windows… but it's not. Wax on windows is bad news, Rain-X is good news. I'm honestly not sure what the difference is, but there definitely is one.
The new stuff cleans as well as adds the protective layer, so you can do it in one operation. I'd recommend, though, that especially for the front windscreen, you repeat the application two or three times over as many days. The older the windscreen - the more motorway miles it has on it - the more this multi-coating is necessary. It lasts maybe three to six months before re-application is necessary, depending on wiper usage. It will also be less likely for car wash wax to stick to it.
You will still see a difference between the wiper-polished & untouched areas of the screen, but it will be less pronounced.
You will also be able to see out at every wipe.