I don’t think resetting the Apple TV will fix a network problem (though there’s always a chance).
Good news:
devices are listed to belong to the same subnet
devices are using the same router / subnet mask
You could double-check the DNS options… though that shouldn’t interfere when it comes only to your local network.
You might want to verify if there’s an option in your router to treat the problematic Apple TV somewhat different (e.g. assigning it to a guest network or other applying other restrictions).
