We have full support for Web Push notifications. I rely on Discourse Chat notifications as part of my job, and they work reliably on my phone and are even mirrored on my smartwatch.
Do you have the instance you use installed as a PWA on your device? That’s step one for heavy usage of an instance, and we prompt every active user for it.
Widgets for PWA are coming in the near future, as the spec is being developed by Microsoft in the Chromium project. I’m keeping an eye for it and will work on it as soon as it’s available as we did with every other PWA feature.