chore: add NewUnstartedHTTPServer helper to disable keep-alives on test servers (#28052)

Tests that proxy or pool connections to a bare `httptest.Server`
intermittently fail on Windows with a bare EOF when a stale pooled
connection is reused. net/http will not retry a non-replayable request
(e.g. a POST) on a closed pooled connection, so forcing a fresh
connection per request eliminates the failure class. This is the same
mechanism fixed in #28016 (AIGOV-430 / internal#1564), now expressed as
a reusable, behavior-preserving helper.

This PR adds `testutil.NewTestHTTPServer`, a known-good wrapper around
`httptest.NewServer` that applies some defaults. Currently the only
default is disabling keep-alives by default.

- `testutil/http_server.go`: `NewHTTPServer(t, handler, opts)` started,
with documented defaults, starts automatically, and handles `t.Cleanup`.
- `testutil/http_server_test.go`: unit test for defaults and overriding
defaults.
- `enterprise/aibridgeproxyd/reload_test.go`: refactor the harness's
hand-rolled server to the helper.

## Future Work

- Functional options are exposed but not explicitly defined. This can be
done later as required.
- No lint rule or broader migration. A forcing-function analyzer
covering more packages, plus wider adoption, belongs in a separate
follow-up.

## Verification

- `testutil` and `enterprise/aibridgeproxyd` suites pass under `-race`.
- `TestProxy_HotReloadRouting` and `TestProxy_StaleTunnel` pass 10x
under `-race`.
- New helper unit test passes under `-race`.

> Generated by a Coder agent.

testutil/http_server_test.go

@@ -0,0 +1,65 @@
+package testutil_test
+
+import (
+	"bufio"
+	"fmt"
+	"net"
+	"net/http"
+	"net/http/httptest"
+	"testing"
+
+	"github.com/stretchr/testify/require"
+
+	"github.com/coder/coder/v2/testutil"
+)
+
+func TestNewUnstartedHTTPServer(t *testing.T) {
+	t.Parallel()
+
+	send := func(conn net.Conn) error {
+		_, err := fmt.Fprintf(conn, "GET / HTTP/1.1\r\nHost: x\r\n\r\n")
+		if err != nil {
+			return err
+		}
+		resp, err := http.ReadResponse(bufio.NewReader(conn), nil)
+		if err != nil {
+			return err
+		}
+		_ = resp.Body.Close()
+		return nil
+	}
+
+	t.Run("defaults", func(t *testing.T) {
+		t.Parallel()
+
+		srv := testutil.NewHTTPTestServer(t, http.HandlerFunc(func(w http.ResponseWriter, _ *http.Request) {
+			w.WriteHeader(http.StatusOK)
+		}))
+
+		conn, err := net.Dial("tcp", srv.Listener.Addr().String())
+		require.NoError(t, err, "dial")
+		defer conn.Close()
+
+		// Keepalives should be disabled: first request succeeds, second request on the same conn fails.
+		require.NoError(t, send(conn), "keepalives disabled: first request on reused connection must succeed")
+		require.Error(t, send(conn), "keepalives disabled: second request on reused connection must fail")
+	})
+
+	t.Run("override", func(t *testing.T) {
+		t.Parallel()
+
+		srv := testutil.NewHTTPTestServer(t, http.HandlerFunc(func(w http.ResponseWriter, _ *http.Request) {
+			w.WriteHeader(http.StatusOK)
+		}), func(srv *httptest.Server) {
+			srv.Config.SetKeepAlivesEnabled(true)
+		})
+
+		conn, err := net.Dial("tcp", srv.Listener.Addr().String())
+		require.NoError(t, err, "dial")
+		defer conn.Close()
+
+		// Keepalives should be enabled: multiple requests on the same conn succeed.
+		require.NoError(t, send(conn), "keepalives enabled: first request on reused connection must succeed")
+		require.NoError(t, send(conn), "keepalives enabled: second request on reused connection must succeed")
+	})
+}