Add E2E tests, CI step that warns developers when a new browser console warning/error appears

[andrii-i]

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Problem

While E2E CI step exists (https://github.com/jupyterlab/jupyter-ai/blob/main/.github/workflows/e2e-tests.yml), there are no functional E2E tests (https://github.com/jupyterlab/jupyter-ai/blob/main/packages/jupyter-ai/ui-tests/tests/jupyter-ai.spec.ts) which can lead to false negatives where CI is green while E2E integration was not actually successfully completed (see #1407, https://github.com/jupyterlab/jupyter-ai/actions/runs/16120919442).

Proposed solution

I think this package could benefit from restoring the browser check which runs a simple playwright runner to check for any errors in console. Ideally having some visual regression tests would be even better, but that would be a first step. A jupyterlab-implemented browser is a part of the extension template, here it is one usage in jupyterlab-git.

Originally posted by @krassowski in #1407