Desktop Safari has had push notifications since like… 10.9 Mavericks. Only mobile Safari doesn't support them.

I thought this too, but it seems Safari only supports the Notifications API for when a site is open, not Web Push for background notifications?

https://caniuse.com/push-api

For Safari specifically you have to use Apple's push notification service (APNS), which is a lot of hassle compared to web push.

There is a ticket open for adding support into Safari, but it probably won't happen anytime soon. https://bugs.webkit.org/show_bug.cgi?id=182566

Wouldn't this be true for any browser? Unless other browsers leave daemons running at all times, which would be kinda creepy.

Most desktop browsers don't run daemons but they do "soft close" for this feature to work. If you completely kill your browser then notifications and service workers stop working.