healthcheck: answer 503 when no root server gave a usable reply in the last round of DNS checks (closes #302)

The health check answered ok while every DNS lookup failed, as on a
network that intercepts DNS queries. A lookup whose root servers give no
usable reply now fails with ErrNoRootServerAnswered. At the end of each
round of DNS checks that shutdown did not cut short, the watcher records
whether every lookup failed with it or with ErrIntercepted; the health
check then gives status error, and /.well-known/healthcheck and /health
answer 503. Until the first round finishes it stays ok.

The health check now depends on the watcher, so fx starts the watcher
before the HTTP server and stops the server first. NewWithRootServers
lets tests start a resolver's walks at real servers that are not root
servers.

Model: opus-5-5

internal/resolver/resolver_test.go

@@ -815,6 +815,7 @@ func TestQueryServers_EveryRootServerRefused(t *testing.T) {
 // but nothing answers at 192.0.2.1, a documentation address, so not
 // every server refused, wherever 192.0.2.1 falls in the random order:
 // the error is not ErrIntercepted and does not say every server refused.
+// No root server gave a usable reply, so it is ErrNoRootServerAnswered.
 func TestQueryServers_NotEveryRootServerRefused(t *testing.T) {
 	t.Parallel()
 
@@ -848,6 +849,34 @@ func TestQueryServers_NotEveryRootServerRefused(t *testing.T) {
 	require.NotErrorIs(t, err, resolver.ErrIntercepted)
 	// Both errors for every server refusing say "refused a query for".
 	require.NotContains(t, err.Error(), "refused a query for")
+	require.ErrorIs(t, err, resolver.ErrNoRootServerAnswered)
+}
+
+// TestLookupsWhenNoRootServerAnswers starts every walk at 192.0.2.1, a
+// documentation address where nothing answers, as if no root server
+// answered. The lookups the watcher makes, LookupNS for a domain and
+// LookupAllRecords for a hostname, both fail with
+// ErrNoRootServerAnswered, which the watcher reads.
+func TestLookupsWhenNoRootServerAnswers(t *testing.T) {
+	t.Parallel()
+
+	r := resolver.NewWithRootServers(slog.Default(), []string{noAnswerAddress})
+
+	t.Run("LookupNS", func(t *testing.T) {
+		t.Parallel()
+
+		_, err := r.LookupNS(t.Context(), "google.com")
+		require.ErrorIs(t, err, resolver.ErrNoRootServerAnswered)
+		require.NotErrorIs(t, err, resolver.ErrIntercepted)
+	})
+
+	t.Run("LookupAllRecords", func(t *testing.T) {
+		t.Parallel()
+
+		_, err := r.LookupAllRecords(t.Context(), "www.google.com")
+		require.ErrorIs(t, err, resolver.ErrNoRootServerAnswered)
+		require.NotErrorIs(t, err, resolver.ErrIntercepted)
+	})
 }
 
 // TestQueryServers_RecursiveResolversAsRootServers passes public