I keep clear of most HP fanfiction, the ones I do read I tell myself 'no, this is fanfic, not actual after the fact.' Cursed Child is a multi-producer play, and the script is more for a visual-based experience. I CAN-NOT believe that some people think actual fanfiction is less fan-ficky than this. Fanfiction is just fiction within fiction, either way and I have seen wackier concepts. So I recommend reading the entire series (with cursed child) before any fanfiction. It gives a better experience.
I definitely agree that the play really read like it was written for the visual experience. From the descriptions I found myself thinking “I would love to see that” because it really seemed like it would look magical in a theater. At this point I think it’s hard to read the full series, including Cursed Child, before reading any fanfic. Many people finished the series 9 years ago and have been reading fanfic ever since to fill the void. I don’t think fanfic is a bad thing and I don’t think reading it before you read Cursed Child affects the experience of reading the play.