backport: Fix the websockets terminate issue

[chenyukang]

What problem does this PR solve?

Problem Summary:

There is a trivial issue introduced in #4314, the shutdown layer didn't applied to websocket after the PR, which makes it will hung up when server terminate websocket.

What is changed and how it works?

Fix the shutdown handler, make it's default for HTTP and Websocket, since we will always send exit signal in these two scenario, so we can make the code simpler.

What's Changed:

Related changes

• PR to update owner/repo:
• Need to cherry-pick to the release branch

Check List

Tests

• Unit test
• Integration test
• Manual test (add detailed scripts or steps below)
• No code ci-runs-only: [ quick_checks,linters ]

Side effects

• Performance regression
• Breaking backward compatibility

Release note

Title Only: Include only the PR title in the release note.