feat(search): one state school group, not academy and council-run

The School type filter split state schools into "academy or free
school" and "council-run". The two were near-halves of one pool (11,186
and 9,316 schools), so choosing one rarely narrowed anything, and the
split did not follow the difference a parent feels most, admissions:
voluntary aided and foundation schools set their own, as academies do.
Faith, which voluntary aided mostly meant, has its own filter.

They are now one group, "State school (free)", leaving five. The old
keys academy and council resolve to state, so a link made with them
keeps working instead of falling through to the raw-label path and
returning nothing.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

backend/app.py

@@ -44,10 +44,10 @@ from .schemas import METRIC_DEFINITIONS, PHASE_GROUPS, RANKING_COLUMNS, SCHOOL_C
 from .school_groups import (
     FAITH_GROUPS,
     FAITH_KEYS,
-    TYPE_GROUP_KEYS,
     TYPE_GROUPS,
     faith_groups_for,
     type_group_for,
+    type_group_key,
 )
 from .nearby_schools import select_nearby
 from .utils import clean_for_json, convert_to_native
@@ -918,17 +918,17 @@ async def get_schools(
             schools_df["local_authority"].str.lower() == local_authority.lower()
         ]
 
-    # A type group key (backend/school_groups.py), or for an old link a raw
-    # GIAS type label, matched exactly as before.
+    # A type group key (backend/school_groups.py), old keys included, or for
+    # an old link a raw GIAS type label, matched exactly as before.
     if school_type:
-        type_key = school_type.lower()
-        if type_key in TYPE_GROUP_KEYS:
+        type_key = type_group_key(school_type)
+        if type_key:
             column = schools_df["school_type"]
             schools_df = schools_df[
                 column.isin(_names_in_group(column, lambda n: type_group_for(n) == type_key))
             ]
         else:
-            schools_df = schools_df[schools_df["school_type"].str.lower() == type_key]
+            schools_df = schools_df[schools_df["school_type"].str.lower() == school_type.lower()]
 
     # Compute result-scoped filter values (before pagination).
     # Gender and admissions are secondary-only filters — scope them to schools