Would it be possible in the future to enable this for the site like YouTube to get notifications easier on our phones. RIght now it's only possible as far as I know to see them by going to the site.

wouldn't that only be possible if the site had an app? other than that you could enable e-mail notifications and do push on your mail app.

It's possible to create a different website if the site is saved on the home screen, maybe push notifs are possible there?

Speedrun.com Notification App would be helpful.