Aside from specific missing apis, the biggest problem with PWAs right now is probably that it's a PITA to maintain local state and also sync it with the server unless you use something like pouchdb that's specifically oriented around that.

Once the filesystem storage apis get worked out, it should be much easier to have persistent sql.js, etc. without having to load the whole thing into memory and save it again constantly and without having to resort to complicated stuff like absurdsql.

I think that finally being able to easily have sqlite in pwas will make it a lot easy to maintain local state and sync it without having to have a maintain a completely separate way of handling data for offline use, ideally without having to rearchitect the whole app as "offline first" but instead just optionally allowing offline use when it's installed as a PWA.

This is assuming you want to enable offline use, but for me that's a major thing that justifies making a page into a PWA (otherwise why not just go to the site?).