feat(frontend): group products by category with ownership filter

Replace category filter dropdown with client-side grouping and a
3-way ownership toggle (All / Owned / Not owned). Products are grouped
by category with header rows as visual dividers, sorted brand → name
within each group. Category column removed (redundant with headings).

Backend: GET /products now returns ProductWithInventory so inventory
data is available for ownership filtering (bulk-loaded in one query).

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

backend/innercontext/api/products.py

@@ -172,7 +172,7 @@ class InventoryUpdate(SQLModel):
 # ---------------------------------------------------------------------------
 
 
-@router.get("", response_model=list[ProductPublic])
+@router.get("", response_model=list[ProductWithInventory])
 def list_products(
     category: Optional[ProductCategory] = None,
     brand: Optional[str] = None,
@@ -205,7 +205,25 @@ def list_products(
             )
         ]
 
-    return products
+    # Bulk-load inventory for all products in one query
+    product_ids = [p.id for p in products]
+    inventory_rows = (
+        session.exec(
+            select(ProductInventory).where(ProductInventory.product_id.in_(product_ids))
+        ).all()
+        if product_ids
+        else []
+    )
+    inv_by_product: dict = {}
+    for inv in inventory_rows:
+        inv_by_product.setdefault(inv.product_id, []).append(inv)
+
+    results = []
+    for p in products:
+        r = ProductWithInventory.model_validate(p, from_attributes=True)
+        r.inventory = inv_by_product.get(p.id, [])
+        results.append(r)
+    return results
 
 
 @router.post("", response_model=ProductPublic, status_code=201)