I just realized that if browsers could display notification badges when a site is closed and showed it on the bookmarked item in the fav bar while the site is not open (no tabs), that'd be super cool and useful! Like I could visibly place sites I commonly expect traffic and allow them to fetch notifications in the background such as this very platform, email, etc. Then I wouldn't need to open the site, "waste" time/resources just to check for new messages and such

hmm, I wonder if RSS is enough for this or if it'd need to be push-based more like XMPP pubsub

Like this? https://developer.mozilla.org/en-US/docs/Web/Progressive_web_apps/How_to/Display_badge_on_app_icon

In the last startup I worked at we built an email inbox meets browser kind of thing, specifically for team collab apps (https://acapela.com). It was not a general solution to the need you're identifying, instead we hardcoded the hell out of it (e.g. re-using users' Notion session cookie to poll their internal API to fetch the notification count)