ci: Install Playwright in the background in more places

[mydea]

Playwright setup takes about 30s in every browser integration, loader and E2E job. In E2E jobs, the existing background install still makes the job wait a median 14s after the app build, which adds up to about 47 runner-minutes per run.

• Split install: the apt install-deps (about 18s) and the browser-cache restore (about 15s) don't depend on each other. A new part input lets the workflow run them as two parallel background steps. background: isn't documented for composite actions, so the split happens in the workflow rather than inside the action. When the cache misses, the browsers part uses its own npm cache, so its npx doesn't race the one installing the system deps.
• Browser integration and loader tests: these now install Playwright while the root node_modules cache restores, instead of afterwards. Background steps must not start next to download-artifact, so restore-cache gets an input to skip its build-output download. The jobs download it before starting the installs.
• E2E: the install now starts right after the artifact downloads, before Install pnpm, Set up Node and the Bun/Deno/SAM setup. Since ci: Read the Playwright version from package.json in install-playwright #25380 it only needs the app's package.json, not any installed dependencies.

Based on #25383, outside the PR stack.

🤖 Generated with Claude Code

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