Only raise integrationnotfound for dependencies (#48241)

Co-authored-by: J. Nick Koston <nick@koston.org>
Co-authored-by: Paulus Schoutsen <paulus@home-assistant.io>

homeassistant/requirements.py

@@ -97,12 +97,21 @@ async def async_get_integration_with_requirements(
             deps_to_check.append(check_domain)
 
     if deps_to_check:
-        await asyncio.gather(
+        results = await asyncio.gather(
             *[
                 async_get_integration_with_requirements(hass, dep, done)
                 for dep in deps_to_check
-            ]
+            ],
+            return_exceptions=True,
         )
+        for result in results:
+            if not isinstance(result, BaseException):
+                continue
+            if not isinstance(result, IntegrationNotFound) or not (
+                not integration.is_built_in
+                and result.domain in integration.after_dependencies
+            ):
+                raise result
 
     cache[domain] = integration
     event.set()