Fix state saving when sharing topics for MQTT entities (#79421)

* Do not write old state sharing availability topic

* Add a test

* Support for all availability topics

* delay async_write_ha_state till last callback

* Process write req after processing callback jobs

* Do not count subscription callbacks

* Simplify

* Stale docsting

* No topic needed for delays state write

* No need to clear when reloading

* Move test to test_mixins.py

* Only set up sensor platform for test

homeassistant/components/mqtt/binary_sensor.py

@@ -47,6 +47,7 @@ from .mixins import (
     warn_for_legacy_schema,
 )
 from .models import MqttValueTemplate
+from .util import get_mqtt_data
 
 _LOGGER = logging.getLogger(__name__)
 
@@ -260,7 +261,7 @@ class MqttBinarySensor(MqttEntity, BinarySensorEntity, RestoreEntity):
                     self.hass, off_delay, off_delay_listener
                 )
 
-            self.async_write_ha_state()
+            get_mqtt_data(self.hass).state_write_requests.write_state_request(self)
 
         self._sub_state = subscription.async_prepare_subscribe_topics(
             self.hass,