[Heartbeat] Cleanup docker install / always add playwright deps (#32122)

Refactors Dockerfile handling of synthetics deps to rely on playwright install-deps rather than us manually keeping up to date with those. This should fix issues with newer playwrights needing additional deps.

This also cleans up the Dockerfile a good amount, and fixes indentation. Finally, this removes the unused Dockerfile.elastic-agent.tmpl file since agent is now its own repo. It also cleans up some other metadata that no longer does anything.

No changelog is specified because no user facing changes are present.