Add backup helper (#139199)

* Add backup helper

* Add hassio to stage 1

* Apply same changes to newly merged `webdav` and `azure_storage` to fix inflight conflict

* Address comments, add tests

---------

Co-authored-by: J. Nick Koston <nick@koston.org>

homeassistant/bootstrap.py

@@ -74,6 +74,7 @@ from .core_config import async_process_ha_core_config
 from .exceptions import HomeAssistantError
 from .helpers import (
     area_registry,
+    backup,
     category_registry,
     config_validation as cv,
     device_registry,
@@ -163,16 +164,6 @@ FRONTEND_INTEGRATIONS = {
     # integrations can be removed and database migration status is
     # visible in frontend
     "frontend",
-    # Hassio is an after dependency of backup, after dependencies
-    # are not promoted from stage 2 to earlier stages, so we need to
-    # add it here. Hassio needs to be setup before backup, otherwise
-    # the backup integration will think we are a container/core install
-    # when using HAOS or Supervised install.
-    "hassio",
-    # Backup is an after dependency of frontend, after dependencies
-    # are not promoted from stage 2 to earlier stages, so we need to
-    # add it here.
-    "backup",
 }
 # Stage 0 is divided into substages. Each substage has a name, a set of integrations and a timeout.
 # The substage containing recorder should have no timeout, as it could cancel a database migration.
@@ -206,6 +197,8 @@ STAGE_1_INTEGRATIONS = {
     "mqtt_eventstream",
     # To provide account link implementations
     "cloud",
+    # Ensure supervisor is available
+    "hassio",
 }
 
 DEFAULT_INTEGRATIONS = {
@@ -905,6 +898,10 @@ async def _async_set_up_integrations(
     if "recorder" in domains_to_setup:
         recorder.async_initialize_recorder(hass)
 
+    # Initialize backup
+    if "backup" in domains_to_setup:
+        backup.async_initialize_backup(hass)
+
     stage_0_and_1_domains: list[tuple[str, set[str], int | None]] = [
         *(
             (name, domain_group & domains_to_setup, timeout)