Review the end-to-end test workflow

[Perdolique]

Context

The build-and-deploy workflow currently runs Playwright end-to-end tests in a dedicated job before deployment. That job installs Chromium and WebKit with their system dependencies and blocks deployment until the test command succeeds.

We want to revisit this setup, but the desired replacement is not decided yet.

Follow-up

• Review whether end-to-end tests should remain a separate blocking job.
• Evaluate reuse of the shared automation workflows and actions.
• Check whether browser installation, caching, sharding, and the browser matrix can be simplified.
• Document the chosen trade-offs before changing the workflow.

This issue intentionally tracks technical debt only; it does not prescribe an implementation.