GNOME and any other desktop environments have a feature of notifications but it doesn’t have a feature of “Push Notifications”.
So you have to always run apps because of receiving notifications. I always run Signal, Thunderbird, Discord, and Microsoft Teams. These need resources of my computer.
However, push notifications will solve that problem. You can receive push notifications even if apps don’t run.
If there is a feature of push notifications integrated with GNOME, developers can make apps with push notifications.
I am not a developer and I don’t know about Desktop Apps well. Please tell me your thoughts.
A standard has already been created for this, the hard part is deploying a push service and keeping it running. I doubt any apps will use it until a standard service becomes stable. A KDE developer did a write-up of this a few months ago: Push Notifications for KDE
Alternatively, it may be better to file bugs with those apps and ask them to support backgrounding, where all the resources from the GUI are freed and only a small service stays around locally to receive notifications. The main reason to use push notifications is so the notifications are not lost when your device goes offline, not necessarily to save resources.