Those technologies are totally independent of each other. The rest API can not be used as it is (in this context I mean the end points) for push notifications as there is a protocol for that and probably all kinds of restrictions browsers place. On the other end, nothing was preventing you from implementing those protocols before there was a rest API, and nothing prevents you from doing it now.
there are even plugins that claim to implement it for years by now https://wordpress.org/plugins/push-notifications-for-wp/ (no idea about the quality of it, was just the first result on google)