[ci] Cache playwright in run_devtools_e2e_tests (#34320)

I happened to notice that I forgot to cache playwright in
run_devtools_e2e_tests, so it would try to install it every time which
can randomly take a while to complete (I'm not sure why it's not
deterministic, but the dependencies appear to be installed
inconsistently across multiple workflows).

This PR adds the same cache we use for other steps that use playwright,
which should shave off some time from this workflow when the cache is
warm.

Additionally I omitted the standalone install-deps command as it appears
to be redundant and adds a lot of extra time to CI, due to the fact that
it installs many unrelated dependencies.

---
[//]: # (BEGIN SAPLING FOOTER)
Stack created with [Sapling](https://sapling-scm.com). Best reviewed
with [ReviewStack](https://reviewstack.dev/facebook/react/pull/34320).
* #34321
* __->__ #34320

.github/workflows/compiler_playground.yml

@@ -57,8 +57,6 @@ jobs:
           key: playwright-browsers-v6-${{ runner.arch }}-${{ runner.os }}-${{ steps.playwright_version.outputs.playwright_version }}
       - run: npx playwright install --with-deps chromium
         if: steps.cache_playwright_browsers.outputs.cache-hit != 'true'
-      - run: npx playwright install-deps
-        if: steps.cache_playwright_browsers.outputs.cache-hit == 'true'
       - run: CI=true yarn test
       - run: ls -R test-results
         if: '!cancelled()'

.github/workflows/runtime_build_and_test.yml

@@ -316,7 +316,7 @@ jobs:
         if: steps.node_modules.outputs.cache-hit != 'true'
       - run: ./scripts/react-compiler/build-compiler.sh && ./scripts/react-compiler/link-compiler.sh
       - run: yarn workspace eslint-plugin-react-hooks test
-        
+
   # ----- BUILD -----
   build_and_lint:
     name: yarn build and lint
@@ -811,9 +811,18 @@ jobs:
           pattern: _build_*
           path: build
           merge-multiple: true
-      - run: |
-          npx playwright install
-          sudo npx playwright install-deps
+      - name: Check Playwright version
+        id: playwright_version
+        run: echo "playwright_version=$(npm ls @playwright/test | grep @playwright | sed 's/.*@//' | head -1)" >> "$GITHUB_OUTPUT"
+      - name: Cache Playwright Browsers for version ${{ steps.playwright_version.outputs.playwright_version }}
+        id: cache_playwright_browsers
+        uses: actions/cache@v4
+        with:
+          path: ~/.cache/ms-playwright
+          key: playwright-browsers-v6-${{ runner.arch }}-${{ runner.os }}-${{ steps.playwright_version.outputs.playwright_version }}
+      - name: Playwright install deps
+        if: steps.cache_playwright_browsers.outputs.cache-hit != 'true'
+        run: npx playwright install --with-deps chromium
       - run: ./scripts/ci/run_devtools_e2e_tests.js
         env:
           RELEASE_CHANNEL: experimental