[v11] Align attaching header data in browser with how it works in Node

Currently, the browser SDK does not collect any headers/cookies when `sendDefaultPii` is `false`.<br><br>In the future, this should be aligned with how it works on the server-side and what the data collection spec describes.<br><br>Related to this PR: [https://github.com/getsentry/sentry-javascript/issues/20706](<https://github.com/getsentry/sentry-javascript/issues/20706>)