Move middleware handling to node server (#33448)

Part of #31506, this PR moves the code of middleware handling from the base server to the node server.

## Bug

- [ ] Related issues linked using `fixes #number`
- [ ] Integration tests added
- [ ] Errors have helpful link attached, see `contributing.md`

## Feature

- [ ] Implements an existing feature request or RFC. Make sure the feature request has been accepted for implementation before opening a PR.
- [ ] Related issues linked using `fixes #number`
- [ ] Integration tests added
- [ ] Documentation added
- [ ] Telemetry added. In case of a feature if it's used or not.
- [ ] Errors have helpful link attached, see `contributing.md`

## Documentation / Examples

- [ ] Make sure the linting passes by running `yarn lint`