[Time box] Migrate E2E tests to playwright (part 2)

## Feature Description

Migrate as many E2E tests as possible to Playwright in the allowed time.

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_Do not alter or remove anything below. The following sections will be managed by moderators only._

## Acceptance criteria

- E2E tests are progressively migrated from Puppeteer to Playwright
- Migrated tests pass in CI via the Playwright runner
- Original Puppeteer specs are removed for each migrated test
- Shared helper utilities are created in Playwright as migration patterns emerge
- Once the time box is reached for this ticket and PR is merged, a follow-up issue is created if further Puppeteer tests remain

## Implementation Brief

- [ ] Work through each remaining `.test.js` E2E tests (until 15 hours of execution time is reached) in `tests/e2e/specs/`, create the Playwright equivalent in the new `tests/playwright` workspace (added in #11912), verify it passes and covers all existing cases, then remove the Puppeteer original:
- Create files `tests/playwright/specs//.spec.js` - rewrite the corresponding test using Playwright APIs and fixtures.
- Remove files `tests/e2e/specs//.test.js` - delete the Puppeteer original once the Playwright spec passes in CI.
- Extract shared utilities as common patterns repeat across migrations.
- **Do not import anything from `tests/e2e` in new playwright specs, create a copy in the playwright folder instead.**

### Test Coverage

* N/A as this issue is testing-related.

## QA Brief

* No QA required as this is an infrastructure-only change.

## Changelog entry

*