chore: update scripts to not use npm lifecycle commands [DX-589]

[chasepoirier]

This pull request updates the release workflow and refactors several npm scripts to improve documentation publishing and test environment setup in response to npm lifecycle scripts being disabled.

Ticket: https://contentful.atlassian.net/browse/DX-589

Updates:

• Added a Publish Docs step to .github/workflows/release.yaml, which automatically publishes documentation when changes are pushed to the master branch.
• Removed the postpublish documentation publishing logic from package.json to centralize publishing in the workflow.
• Refactored the browser integration test setup in both package.json and test/output-integration/browser/package.json