Fix integration test failures in CI

MR: Pending

Description

This is a follow up issue to investigate and fix integration test failures in CI. We're currently ignoring these failures by adding the allow_failures GitLab CI configuration: chore: Allow integration test failures (!504 - merged).

Failing pipeline: https://gitlab.com/gitlab-org/gitlab-web-ide/-/jobs/10300028111

Playwright report: https://gitlab-org.gitlab.io/-/gitlab-web-ide/-/jobs/10300028111/artifacts/playwright-report/index.html

Acceptance criteria

• Investigate and fix the cause of flakey tests when running in CI
• Once fixed, remove allow_failure from GitLab CI config