Just some anecdata: I use the Outlook PWA (on an iPhone!) to access my work mail and calendar. It does not have notifications, which I really liked for calendar events (because I’m forgetful), so that’s a bit sad. However, there is a decisive pro: It cannot enforce restrictive device policies. What a great feature!
UX isn’t half bad either. It actually feels pretty native most of the time.
On the other hand our IT recently blocked external sharing of calendars, which I'd used to gain read-only access to my work calendar on my private phone and I bloody hate it – by the time the browser and Outlook have loaded on my under-powered phone, I could have launched my calendar app ten times over, plus the work calendar is now no longer integrated with the system calendar APIs so it no longer appears on my homescreen calendar widget, I now I have to switch back and forth between separate apps for my private and my work calendar, offline access only works so-so and wastes additional space, etc. etc.
(Plus I don't want to actually deal with full Outlook because I have zero interest in accessing my work mail privately, whereas I do need to take a look at my work calendar in order to avoid accidentally double-booking myself.)
I would certainly think so? The PWA currently does not contain anything about notifications.
Even on your run-off-the-mill desktop web browser, push notifications do not come “for free”. You need a service worker. You need a specific server implementation. It’s not just the Notification API.
UX isn’t half bad either. It actually feels pretty native most of the time.