I agree 100% with sunshine wr, a good machine buff with
cerium oxide is probably the best and safest solution to your problem.
As for the cause of the problem, I think it is likely a coating on the windshield, which may have been deliberate, or may be just a spray wax or some other chemical that many automatic car washes use these days as their final step.
Wax, polish, or hydrophobic coatings like Rain-X or Aquapel leave a transparent film on the glass that helps repel water and contaminants. In some cases they also make the wipers work more efficiently but as they begin to break down the film becomes more visible, particularly just after using the wipers to remove water from the glass. The spots and streaks often occur on coated windshields when the vehicle is left outside in the rain and then allowed to air dry, and again, is much more apparent when you start using your wipers but may be nearly invisible when the windshield is wet or completely dry.
Sometimes simply applying a fresh application of a good hydrophobic coating will take care of the problem but that may only work if you know what was used previously and if the windshield is cleaned well before the coating is applied.
Depending on what was used on the windshield removing the coating may be as simple as using wax and adhesive remover, or pumice mixed with water to remove it. Hand applied
Creme Cleanser is what I use and for tempered side and back glass I don't think anything works better but it is pretty aggressive so for softer glass like mirrors and windshields it can leave very fine scratches in the glass if you are not very careful. Actually I don't think it is the cleanser that leaves the scratches but as it dislodges the mineral deposits that are often the cause of water spotting and that my cause scratching. So for windshields and mirrors I agree 100% with sunshine wr, a good machine buff with cerium oxide is likely your best option.
I also recommend that you clean your wiper blades thoroughly with denatured alcohol or replace them at the same time you remove any coatings from your windshield. Wiper blades pick up trace amounts of the coating from your windshield with every pass and may continue to leave marks if they are not cleaned or replaced.
I have a love/hate relationship with hydrophobic coatings. I really hate them when I forget to test for them prior to starting a windshield repair and only realize it when my pit resin does not stick to the glass or when it starts to wear off and I experience the problem you are having. However, I love how well I can see in a hard rain at highway speeds without even turning on the wipers, I love how it reduces hard water spotting (especially on my mirrors and on my side and back glass with privacy tint), and I love how much easier it is to clean bugs off the windshield.
I apply Rain-X once a year to my mirrors, side, and back glass, but every 2-4 months on my windshield depending on weather conditions and find with that frequency I do not have to remove the previous application, I just have to make sure the glass is cleaned well. I also clean my wiper blades every time I apply a coating and replace them every year or two.