In almost every case like this, the Apple TV is not starting playback by itself, something else in the home is telling it to play, and the Podcasts app is just the last thing that was queued. Test the suspects in this order:
1. AirPlay from another device. On the Apple TV go to Settings > AirPlay and HomeKit and set Allow Access to Only People Sharing This Home, or turn AirPlay off entirely for one night. If the phantom playback stops, an iPhone, iPad, Mac, or HomePod on your network was sending it. The usual sources: a Shortcuts automation on an iPhone set to play podcasts at a time of day and routed to the TV, or a phone that resumes Podcasts (CarPlay and Bluetooth connections do this) with the Apple TV still selected as its output.
2. Check the Podcasts app on your iPhone: if the episodes it plays at night are also progressing in your Up Next there, the start is coming from the phone side, and the fix lives there, not on the TV.
3. HDMI-CEC. A TV or receiver can wake the Apple TV and effectively press play when it powers up or switches inputs. Test by turning off Settings > Remotes and Devices > Control TVs and Receivers for a night or two (you lose power and volume sync while testing). Some TVs also have their own resume last content setting worth disabling.
The unmuting fits this picture: whoever or whatever starts the playback brings its own volume with it, so the mute you set on the TV does not survive. Start with the AirPlay test tonight, it rules the biggest suspect in or out in one step.
