Restore states through a JSON store instead of recorder (#17270)

* Restore states through a JSON store

* Accept entity_id directly in restore state helper

* Keep states stored between runs for a limited time

* Remove warning

homeassistant/components/binary_sensor/mqtt.py

@@ -102,8 +102,7 @@ class MqttBinarySensor(MqttAvailability, MqttDiscoveryUpdate,
 
     async def async_added_to_hass(self):
         """Subscribe mqtt events."""
-        await MqttAvailability.async_added_to_hass(self)
-        await MqttDiscoveryUpdate.async_added_to_hass(self)
+        await super().async_added_to_hass()
         await self._subscribe_topics()
 
     async def discovery_update(self, discovery_payload):