[Bug]: trace not recorded for "workerStorageState"

@dgozman is already working on this.

Since Jun 24, 2024.

### Version

18.2.3

### Steps to reproduce

1. clone my repo - https://github.com/Danieliverant/playwright-no-trace
2. `cd playwright-no-trace`
3. `nx e2e e2e --trace on`
4. `npx playwright show-trace ./e2e/playwright-report/example-has-title-chromium/trace.zip`

(the test will fail)

### Expected behavior

To see the steps in the "wrokerStorageState"

### Actual behavior

Not seeing anything from the "wrokerStorageState"

### Additional context

Same thing for UI mode - you wait until the "workerStorageState" to finish and only then you see the trace.

### Environment

```shell
System:
OS: macOS 13.6
CPU: (10) arm64 Apple M1 Pro
Memory: 85.05 MB / 16.00 GB
Binaries:
Node: 18.10.0 - ~/.nvm/versions/node/v18.10.0/bin/node
npm: 8.19.2 - ~/.nvm/versions/node/v18.10.0/bin/npm
Languages:
Bash: 3.2.57 - /bin/bash
npmPackages:
@playwright/test: ^1.36.0 => 1.43.0
```