Revert "fix(mcp): drop the backend after browser_close with a shared browser"

[Yury Semikhatsky (yury-s)]

Reverts #42365

[github-actions]

Test results for "MCP"

1 failed
❌ [firefox] › mcp/cli-mouse.spec.ts:38 › mousewheel @mcp-windows-latest-firefox

8261 passed, 1371 skipped

---

Merge workflow run.

[github-actions]

🟢 CI is clear — the one failure is a pre-existing flake

Hi, I'm the Playwright bot and I triaged the failing CI for this PR.

The only failure — [firefox] › mcp/cli-mouse.spec.ts:38 › mousewheel on mcp-windows-latest-firefox — is a known cross-run flake, not something this revert introduced. This PR only reverts the MCP backend lifecycle changes from #42365 (browserBackend.ts / mcp/index.ts and their tests); it doesn't touch mouse input, so nothing in the diff reaches this test.

Details

Pre-existing flake / infra

• [firefox] › mcp/cli-mouse.spec.ts:38 › mousewheel (mcp-windows-latest-firefox) — this test flips verdict across runs on that exact bot: passed 689 of 697 runs, failed 8. The failures land on SHAs this PR can't be responsible for, including other PRs (fix(tracing): allow restarting tracing after a failed stop #42425, docs(browsercontext): undeprecate setHTTPCredentials #42419, feat(chromium): expose tab strip metadata via page.tabInfo() #42355, feat(codegen): add --http-credentials option #42322, chore(expect): pass expect title explicitly to library calls #42250) and plain push runs on main. It's Windows-Firefox-only and fails on the first attempt with no retry rescue, so the signature is consistent with the known flake rather than this change. The diff here is an MCP backend-close revert, which the mousewheel path doesn't exercise.

_{Triaged by the Playwright bot - agent run}