Refactor integration startup time tracking to reduce overhead (#110136)

* Refactor integration startup time tracking to reduce overhead

- Use monotonic time for watching integration startup time as it avoids incorrect values if time moves backwards because of ntp during startup and reduces many time conversions since we want durations in seconds and not local time

- Use loop scheduling instead of a task

- Moves all the dispatcher logic into the new _WatchPendingSetups

* websocket as well

* tweaks

* simplify logic

* preserve logic

* preserve logic

* lint

* adjust

tests/test_setup.py

@@ -1,7 +1,6 @@
 """Test component/platform setup."""
 
 import asyncio
-import datetime
 import threading
 from unittest.mock import AsyncMock, Mock, patch
 
@@ -714,28 +713,22 @@ async def test_integration_only_setup_entry(hass: HomeAssistant) -> None:
 async def test_async_start_setup(hass: HomeAssistant) -> None:
     """Test setup started context manager keeps track of setup times."""
     with setup.async_start_setup(hass, ["august"]):
-        assert isinstance(
-            hass.data[setup.DATA_SETUP_STARTED]["august"], datetime.datetime
-        )
+        assert isinstance(hass.data[setup.DATA_SETUP_STARTED]["august"], float)
         with setup.async_start_setup(hass, ["august"]):
-            assert isinstance(
-                hass.data[setup.DATA_SETUP_STARTED]["august_2"], datetime.datetime
-            )
+            assert isinstance(hass.data[setup.DATA_SETUP_STARTED]["august_2"], float)
 
     assert "august" not in hass.data[setup.DATA_SETUP_STARTED]
-    assert isinstance(hass.data[setup.DATA_SETUP_TIME]["august"], datetime.timedelta)
+    assert isinstance(hass.data[setup.DATA_SETUP_TIME]["august"], float)
     assert "august_2" not in hass.data[setup.DATA_SETUP_TIME]
 
 
 async def test_async_start_setup_platforms(hass: HomeAssistant) -> None:
     """Test setup started context manager keeps track of setup times for platforms."""
     with setup.async_start_setup(hass, ["august.sensor"]):
-        assert isinstance(
-            hass.data[setup.DATA_SETUP_STARTED]["august.sensor"], datetime.datetime
-        )
+        assert isinstance(hass.data[setup.DATA_SETUP_STARTED]["august.sensor"], float)
 
     assert "august" not in hass.data[setup.DATA_SETUP_STARTED]
-    assert isinstance(hass.data[setup.DATA_SETUP_TIME]["august"], datetime.timedelta)
+    assert isinstance(hass.data[setup.DATA_SETUP_TIME]["august"], float)
     assert "sensor" not in hass.data[setup.DATA_SETUP_TIME]