What makes you believe that ?
It uses browser storage. In the absolute, web and local storage store infos at the same place in the browser (not the cache, the storage, a dedicated space for those infos thanks to HTML5 specs).
It all uses the storage of the browser ,but it uses different kind to store the data. LocalStorage uses the IndexedDB, WebStorage uses local storage and session storage.
Moreover, it shouldn't solve the problem of storage files being cleared by the OS.
I don't think using webstorage would help with the issue at hands.
I found out that iOS 8 and 9 has bugs in IndexedDB and WebSQL. So my suggestion was to try WebStorage, because I didn't find a bug report about local storage on iOS 9. I only suggest that he could try, I didn't say that is would fix the problem.
If you could somehow create files on the system and read them, that could be something considered, but I doubt this will be an easy treat, especially on Apple device.
Shaming them for their awful software and asking them to comply better with HTML5 specs could be the only solution around.
But they don't seem to care a lot as per they slow release cycle which goes on breaking previous features.
It is a shame, the bugs are quite old. But it looks Apple don't care about.