fix(search): offer every filter option, not only those in the results

School type, gender and admissions took their options from the result
set, which the filter had already narrowed: choose "Girls" and only
"Girls" was offered, so switching to "Boys" meant clearing first. They
now offer the full lists from /api/filters, as phase already did. Local
authority stays scoped to the results, so a postcode search offers the
councils nearby rather than all 153.

Whether gender, sixth form and admissions show was also decided by the
results (any secondary school in them). It is now decided by the phase
alone: hidden for Primary, Nursery and Middle deemed primary, shown
otherwise. Choosing one of those phases clears the three filters, which
would otherwise stay applied with no control showing.

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

e2e/tests/journeys.spec.ts

@@ -286,6 +286,23 @@ test('the phase filter switches straight from secondary to primary', async ({ pa
   await expect(phase).toHaveValue('primary');
 });
 
+test('school type and gender switch straight to another value', async ({ page }) => {
+  // Their options came from the result set, which the filter had already
+  // narrowed, so with one value chosen it was the only one on offer.
+  await page.goto('/?search=school&gender=girls');
+  // An applied gender filter opens More filters by itself.
+  const gender = page.getByRole('combobox', { name: 'Gender', exact: true });
+  await expect(gender).toHaveValue('girls', { timeout: 15_000 });
+  await gender.selectOption('boys');
+  await expect(page).toHaveURL(/[?&]gender=boys(&|$)/);
+
+  const type = page.getByRole('combobox', { name: 'School type', exact: true });
+  const [first] = (await type.locator('option').allTextContents()).slice(1);
+  await type.selectOption(first);
+  await expect(page).toHaveURL(/[?&]school_type=/);
+  await expect.poll(() => type.locator('option').count()).toBeGreaterThan(2);
+});
+
 test('a phase outside primary/secondary filters to that phase, not to everything', async ({ page }) => {
   // The search page offers every GIAS phase, but the API only knew the grouped
   // ones and silently dropped the rest — so "Nursery" returned primaries.