You would not; you must grant a permission to push notifications to the site first.

I'm grateful I don't get random popovers asking me whether or not I would approve a popup asking me to grant permissions for push notifications (because the scripts exit early on mobile safari)

I don't feel overwhelmed by popups asking me to grant permissions for push notifications. But I'm glad that both sites where I care for them, can push them.

I mean, yeah, with a thing popping up on my screen to ask me for it. Which is fine if I, say, install it as an app (eg PWA or whatever), but super annoying on websites I’m visiting for the first time.

If other users were able to get notifications from websites that doesn't mean that you would get them, especially not from random websites.