v2.338.0: steps that start a background server and poll localhost fail immediately ("Socket closed before the connection was established"); v2.337.0 is fine #4788

Describe the bug
After our self-hosted runners moved from v2.337.0 to v2.338.0, jobs that start a
dev server in the background and then wait for it on localhost started failing
within about a second. The same repo code and workflow pass on v2.337.0.

The failing step uses a Cypress GitHub Action that runs
start: <dev server command> and wait-on: http://localhost:<port>. The log
shows "waiting on http://localhost:" and then immediately:

##[error]Socket closed before the connection was established

The step previously took several minutes (server start plus tests). Nothing from
the dev server is logged before the failure.

What is and isn't affected

• Build, lint, unit-test and type-check jobs on the same runners pass.
• Only steps that start a background server and connect to localhost fail.
• Reproduced in two unrelated repos, using two different versions of the same
  Cypress action (v2 and v7.3.0), different package managers (yarn, pnpm) and
  different Node versions (20.20.0, 22.16.0).

To Reproduce

1. Self-hosted Linux x64 runner, ephemeral, on v2.338.0.
2. Run a workflow step that starts a dev server in the background and waits on
   http://localhost:<port> (we used a Cypress action with start and wait-on).
3. The step fails in about 1 second with the error above.

Expected behavior
Same as v2.337.0: the server starts, wait-on connects, tests run.

Evidence that it's the runner version

• Last passing run: runner 2.337.0. First failing runs: runner 2.338.0.
• No change to the workflow or repo code between them.
• Rolling the runner back to v2.337.0 fixed it. Re-running the same failed jobs
  now passes.

Possibly related changes in v2.338.0 (unconfirmed)

• feat(runner): use node24 for internal operations #4752 internal Node moved to Node 24 (hashFiles, --check, runsvc.sh)
• Implement broker-listener WebSocket connectivity probes. #4753 broker-listener WebSocket connectivity probes

We haven't identified the mechanism. Happy to gather runner _diag logs or try
ACTIONS_RUNNER_FORCED_INTERNAL_NODE_VERSION=node20 if that helps.

Runner Version and Platform

Version of your runner? 2.338.0 (fails), 2.337.0 (works)
OS of the machine running the runner? Ubuntu 22.04 (AWS EC2, x64, ephemeral runners)

Job Log Output

waiting on "http://localhost:<port>" with timeout of 180 seconds
##[error]Socket closed before the connection was established