Twitter had some great outcomes when they rolled out their PWA: https://web.dev/case-studies/twitter
Twitter Lite is now the fastest, least expensive, and most reliable way to use Twitter. The web app rivals the performance of our native apps but requires less than 3% of the device storage space compared to Twitter for Android.
65% increase in pages per session
75% increase in Tweets sent
20% decrease in bounce rate
yet they kept pushing their native apps, probably because they can collect more data through them. The web is way more sandboxed than regular apps.
Web apps have supported push notifications for a long time now. I think even Safari supports them now.