Discourse does support mobile push notifications actually, but iOS does not allow web push notifications through PWAs (which is how most probably install Discourse on mobile). If you use discourse hub on iOS, you do get push notifications.
Many of us internally use DiscourseHub for Meta and our internal instance. Some people like to keep Discourse push notifications muted by disabling push notifications to DiscourseHub. This is an issue when chat is introduced because everyone wanted chat push notifications. The easiest solution for this, is the setting you screenshot. It still allows chat push notifications through while blocking all core push notifications.