You do you! Instagram works for some threat models. The fact that you’re thinking about containing this in the form of a web app is a positive step!
Discourse forums are PWAs, though I’m not sure how they show in iOS. Generally, a PWA will show in its own window like a native app and sometimes also send notifications like a native app.
Besides that, I would check cookie settings to see what the deal is. That’s all I can think off because I stay logged into my PWAs whether I log in from the browser or after I install the app.