fix(mcp): do not run heartbeat for clients without the event stream

[Yury Semikhatsky (yury-s)]

Summary

• The streamable HTTP transport silently drops server->client requests until the client opens the GET event stream, which is optional per spec. A POST-only client could never answer the heartbeat ping, so its session was reaped ~5s after the first tool call.
• Start the heartbeat only once the event stream is open, at most once per server. The 5s transport readiness cap now applies to the roots listing only.

Fixes #42188

[github-actions]

Test results for "MCP"

2 failed
❌ [firefox] › mcp/annotate.spec.ts:269 › should start dashboard and annotate when no dashboard is running @mcp-windows-latest-firefox
❌ [firefox] › mcp/cli-core.spec.ts:57 › click link @mcp-windows-latest-firefox

8087 passed, 1284 skipped

---

Merge workflow run.

[github-actions]

Hi, I'm the Playwright bot and I took a look at the CI failures here.

🟢 Both failures are pre-existing flakes — this PR is clear

mcp/annotate.spec.ts:269 › should start dashboard and annotate when no dashboard is running and mcp/cli-core.spec.ts:57 › click link both flake on Firefox with the identical signature on many unrelated PRs. This PR only defers the MCP HTTP streamable heartbeat, which neither the annotate dashboard nor the stdio CLI flow exercises.

Details

Overall: no failure traces back to this change. The diff is confined to the MCP HTTP streamable transport — mcp/http.ts and mcp/server.ts (starting the heartbeat only once the GET event stream opens) plus a new tests/mcp/http.spec.ts case. Neither failing test goes through that transport or the heartbeat.

Pre-existing flake / infra

• [firefox] › mcp/cli-core.spec.ts:57 › click link (@mcp-windows-latest-firefox) — pre-existing flake. In the aggregated test-results DB this test is bimodal on Firefox: failed 9 of 709 Firefox runs, passed 700 (0 failures on chromium/webkit). Every failure carries the same signature — expect(received).toContain(expected) missing the ### Page snapshot. It failed the same way on unrelated PRs fix(cli): hide Windows daemon consoles #42155 (9fa4010d), feat(chromium): roll to r1237 #42072 (6f00d1da), test(page): cover file URL to about:blank navigation #42053 (377ae5dd), chore: mark v1.62.1 #42020, fix(aria): preserve names from collapsed text contributors #41988 and others — SHAs this PR can't be responsible for → flake.

• [firefox] › mcp/annotate.spec.ts:269 › should start dashboard and annotate when no dashboard is running (@mcp-windows-latest-firefox) — pre-existing flake. Bimodal and cross-browser: failed 1/717 chromium, 1/724 chrome, and a handful of Firefox runs, passing the rest; every failure is the same expect(locator).toBeVisible() on getByRole('main', { name: 'Dashboard' }). Same failure on unrelated PR feat(trace): add action screenshot and aria snapshot events #42167 (4c352624), PR feat(webkit): roll to r2339 #41984 (e17cb8b8), fix(registry): evaluate defaultCacheDirectory lazily #41942 (dc98c5cf), and pushes to main (19630696, 83768264) → flake.

_{Triaged by the Playwright bot - agent run}