I think it's reasonable for Zulip to ask for compensation for access to these gateways, since Apple and Google do not make them available to end users free of charge, and the burden of responsibility to ensure that these systems aren't abused is on them. Also, the fact that they offer mobile push notifications for any self hosted server of up to 10 users is pretty generous, and there seems to be a Community plan option for larger servers that includes "groups of friends" as a qualifier. It really seems they're offering quite a bit.
On its own notification to your device will happen eventually when the ntfy app on your phone wakes up and polls. Pull, not push.
My ntfy server has a config line for an upstream, which is a service that then uses push. Basically it’s self hosted and handing off push.
https://docs.ntfy.sh/subscribe/phone/#instant-delivery
The upstream approach you describe is only necessary for iOS devices that don't permit apps to do that.
But I really wish Android supported specifying additional servers to poll (and/or replace the default server), so you could use a self-hosted service in addition to or instead of Google's service.