[Feature]: Attach webServer output logs as whole run artifacts

### 🚀 Feature Request

When webServer exits prematurely you only get to see the error message in report but not the actual logs. Even if you pipe stdout and stderr in the config. One option is to check ci/cd logs but with all other things happening there it's not always convenient.

It would be useful, in case stdout and stderr is piped, that those logs would be attached to whole run (not some specific test). Perhaps it could be configurable if one does not want those logs.

### Example

_No response_

### Motivation

-

Start by tracing how webServer stdout and stderr are piped and how whole-run artifacts are currently attached. Determine the existing handling for prematurely exiting webServer processes and identify the relevant tests for reports or artifacts. Done means configured logs are attached to the whole run rather than a specific test, with optional behavior if supported.

Written by the indexing model from the issue text.