Seconding the PWAs here, but I caution against thinking of it as an "either-or" situation.
If you're going to build an app that can be accessed in a browser from a url, there are only benefits to making it a PWA. Offline, local capabilities, wider native device access, first-class icon placement with mobile apps, and the simplest distribution network ever invented. Absolute win to make your app a PWA.
That said, it may lack some polish? I never had any issue with what they offered because I HATE apps. I will NEVER install an app on my device if I can help it. No facebook, no twitter, no amazon. If I should be able to do everything the app needs with a browser, then I will NOT download that app. Twitter and amazon at least understand that enough to have functional websites on all platforms. Things like facebook, on the other hand, are 'never installs' for me. So if there's something that capital-A Apps provide that you need, then you should probably go ahead and make one (hopefully using the same UI code you use for your PWA/web-app, but that's not always possible). Things like Twitch and YouTube get apps because they deal with a very specific type of data and can be (and are) optimized to handle that specific data type. Browsers just don't serve that kind of special-purpose design (nor should then), so there's valid reasons for those to be apps.
On the other hand, if you can avoid making a single-purpose app, you should probably try it? Besides the wonks like me refusing app downloads, you also have the headache of maintenance for multiple platforms and all of that pretty much necessitates code mutations and generations for each specific platform that then each all need to be maintained (or worse, in-house developed and maintained). For all of the people out there pitching "holy grail" versions of the write-once, publish-everywhere mantra, I haven't found any that don't include some pretty fine text to the tune of "we use these specific processes and you've just gotta roll with that, even though they will probably cause you compatibility issues in your codebase at least once"
As far as storage, though, localStorage is pretty sparse, it's true. But you can also use IndexedDB storage (pretty good!), and now there's a new file system api that mimics a disk file system in the browser, which is apparently way better but I don't have any first hand experience with that (and it's not useful to me until it can act like a database, but SQLite does seem to be working dilligently to make that happen ASAP). So there are options there, and while people may waffle about the reliability of some of them, I personally haven't run into any issues using IndexedDB (other than it's atrocious development experience).