Improve Playwright test patterns in providers.spec.ts

[hussain-s]

Summary

Improve Playwright test patterns in providers.spec.ts and ProvidersPage.ts without changing test coverage or behavior.

What changed

• replaced DOM-query based waiting with locator-based readiness checks
• moved Admin menu navigation into the page object
• replaced manual row-count and string assertions with web-first Playwright assertions
• kept the existing providers test coverage intact

Validation

• pnpm exec eslint tests/e2e/specs/providers.spec.ts tests/e2e/pages/ProvidersPage.ts
• pnpm exec tsc --noEmit -p tsconfig.app.json
• TEST_USERNAME=admin TEST_PASSWORD=admin AIRFLOW_UI_BASE_URL=http://localhost:28080 pnpm exec playwright test tests/e2e/specs/providers.spec.ts --project=chromium

Relates to #63431

[boring-cyborg]

Congratulations on your first Pull Request and welcome to the Apache Airflow community! If you have any issues or are unsure about any anything please check our Contributors' Guide (https://github.com/apache/airflow/blob/main/contributing-docs/README.rst)
Here are some useful points:

• Pay attention to the quality of your code (ruff, mypy and type annotations). Our prek-hooks will help you with that.
• In case of a new feature add useful documentation (in docstrings or in docs/ directory). Adding a new operator? Check this short guide Consider adding an example DAG that shows how users should use it.
• Consider using Breeze environment for testing locally, it's a heavy docker but it ships with a working Airflow and a lot of integrations.
• Be patient and persistent. It might take some time to get a review or get the final approval from Committers.
• Please follow ASF Code of Conduct for all communication including (but not limited to) comments on Pull Requests, Mailing list and Slack.
• Be sure to read the Airflow Coding style.
• Always keep your Pull Requests rebased, otherwise your build might fail due to changes not related to your commits.
  Apache Airflow is a community-driven project and together we are making it better 🚀.
  In case of doubts contact the developers at:
  Mailing List: dev@airflow.apache.org
  Slack: https://s.apache.org/airflow-slack

[choo121600]

Thanks for Contribution!
LGTM :)

[boring-cyborg]

Awesome work, congrats on your first merged pull request! You are invited to check our Issue Tracker for additional contributions.