I think push notifications are a very important part of an OS to be more usable for a broader range of users.
With UnifiedPush, work has been done to create a specification and tools to easily setup push notifications without relying on a proprietary centralized service.
You mean, to trigger notifications on a Linux system when a connected device (smartphone) gets notifications?
That’s not the scope of UnifiedPush unfortunately - It’s more or less to handle push notifications on mobile devices (PureOS on Librem 5, Mobian, Plasma Mobile, PostmarketOS, …) directly