Using what is provided by mobile operating systems is the standard. FastMail is decoupling JMAP from notifications because it's a sound architectural decision for that reason.

If you want good battery life, there is no alternative to the mobile notifications provided by the platform, because otherwise you have to always keep your own connection open to your server.

please don't spread misinformation - abusing long live socket connections is what drain the battery - having one open and managing it in sane manner is quite battery efficient. Google brainwashed everyone to force them to use their push service, which is just plain awful... (and requires open connection... sic!)

That's simply not true if implemented correctly. Maintaining an open connection yourself has at most minimal impact on battery life. See for example Conversations (XMPP client) which is able to do this just fine on Android.