You can bank without Apple iOS/macOS or Google Android/Chrome.

You can also vote with your wallet regarding banking. Put your deposits in another bank.

GrapheneOS accepts donations and, to my knowledge, they spend it in hiring full time engineers.

They have replacement portions for Play Services already (attestation, an app store, and location), but it'd be interesting if they also offered something for push notifications.

If you have it to give you can spend your budget on a donation and fund the effort directly.

I believe GrapheneOS talked about hosting their own unifiedpush server and baking it into the OS not long ago

Some apps offer direct APK downloads from their websites. If Google Play Services is detected, they use it for push notifications. Otherwise, they fall back to an internal background connection. WhatsApp is an example.

You still need to supply Play Services:

GrapheneOS uses Sandboxed Google Play. LineageOS requires flashing a package like MindTheGapps. There is also microG, an open-source reimplementation of Play Services APIs.