Thunderbird for Android must keep running in the background to maintain IMAP IDLE/push connections. On many devices, the usual workaround is to keep the app in the recent apps (overview) screen and lock its card so it can't be removed by the "Clear all" button. This has two downsides:
Add an opt-in setting, for example under Settings → General settings → Privacy, called "Hide from recent apps" (default: off).
When enabled, Thunderbird's task is excluded from the recent apps screen via android:excludeFromRecents / FLAG_ACTIVITY_EXCLUDE_FROM_RECENTS. This only removes the task card from the overview. It does not stop the app. With push enabled, the existing PushService foreground service (behind the ongoing "Waiting for new emails" notification) keeps running and keeps the IMAP IDLE connections alive. The app doesn't need to be in recents to receive mail, and tapping the launcher icon or a notification still opens it normally.
With no card in recents, there is nothing to swipe away by accident, which is what users who rely on push/IDLE want. Users who prefer the current behavior can leave the option off.
#5938 requests FLAG_SECURE to block screenshots. That only blanks the preview. The task card stays in recents (the issue itself notes the app remains visible in the list of open apps). #6560 was closed as a duplicate of #5938, but its "hide content in the multitasking view" use case is only partly addressed by that approach.
This request is about removing the card entirely and should not block screenshots.
Several Android apps already offer a comparable "hide from recents" option (e.g. Clash Meta, AccessibilityManager). This is a self-contained setting that would make the push/IDLE workflow much cleaner.
I'm happy to test a build or provide more details.