Yep, I've come to the same conclusion. It doesn't seem possible to persist data on itch on iOS or Safari due to the iframe restriction. When I uploaded the same exact build of my game to my own site for testing, the data does persist (using IndexedDB storage).
I briefly tried experimenting with some Unity javascript plugin code to read & write cookies (and also tried window.localStorage instead) but neither one seemed to behave any different and also didn't persist data.
IF @Leafo can offer any help or insights here that would be amazing!