One diagnostic step that narrows this down fast, because these playback cards can come from three different sources and each has its own off switch: next time the popup appears, long press it. The expanded view names what generated it, and the options there let you turn off notifications for that source directly.
The three candidates, in the order I would check them:
1. The Home app. If your Apple TVs are accessories in your Home, their playback status can arrive as Home activity. Open Home, go to Home Settings, then Notifications, and look for the TVs under status and activity notifications, or open the Apple TV accessory's own settings and turn its notifications off there.
2. The TV app's notification channel in Settings > Notifications > TV, separate from the Live Activities switch you already turned off. Banners and Lock Screen alerts for the TV app live here.
3. If the header on the long press shows it is the system media/Live Activity surface rather than an app, there is no per app switch left to flip, which points at the version mismatch you already spotted: the phones moved to 27.2 and the Apple TVs are still on 26, and mixed versions are when these cross-device surfaces tend to misbehave. In that case the long press's Turn Off option for that notification source is the only clean stop, and it is worth a Feedback report since every documented toggle is already off on your devices.
