Add a filter action which would allow you to send a UnifiedPush notification when the filter criteria have been met. UnifiedPush is a decentralized push notification system that lets you choose the service you want to use. It’s designed to be privacy-friendly, flexible, and open - see https://unifiedpush.org. I use the NTFY app on my de-Googled GrapheneOS phone for push notifications from most of my apps.

Being able to write e-mail filter(s) that send me a push notification (without involving Google or Google Play Services) would be very helpful in handling my e-mail inbox. ProtonMail only has Google-based push messages for ALL incoming messages which lacks both privacy and user control.