chore(proxy): drop legacy 'per-context' launch proxy handling

[Yury Semikhatsky (yury-s)]

Summary

• per-context launch proxy was a placeholder for an old Chromium workaround (removed in chore: remove Chromium Windows proxy hacks #31724); treat it as no launch proxy at launch time.
• Remove the per-context string checks in fetch and client certificates, which never matched the normalized http://per-context value.

[github-actions]

Test results for "tests 1"

8 flaky

⚠️ [chromium-library] › library/browsercontext-page-event.spec.ts:173 › should work with Ctrl-clicking `@frozen-time-library-chromium-linux`
⚠️ [chromium-library] › library/video.spec.ts:356 › screencast › should work for popups `@frozen-time-library-chromium-linux`
⚠️ [chromium-library] › library/video.spec.ts:762 › screencast › should work with video+trace `@chromium-ubuntu-22.04-arm-node20`
⚠️ [chromium-library] › library/video.spec.ts:725 › screencast › should capture full viewport on hidpi `@chromium-ubuntu-22.04-node24`
⚠️ [chromium-library] › library/video.spec.ts:113 › screencast › should capture static page `@chromium-ubuntu-22.04-node20`
⚠️ [chromium-library] › library/video.spec.ts:762 › screencast › should work with video+trace `@chromium-ubuntu-22.04-node20`
⚠️ [firefox-library] › library/browsercontext-cookies-third-party.spec.ts:257 › third party 'Partitioned;' cookies `@firefox-ubuntu-22.04-node20`
⚠️ [firefox-library] › library/browsercontext-cookies-third-party.spec.ts:470 › top level 'Partitioned;' cookie and same origin iframe `@firefox-ubuntu-22.04-node20`

52144 passed, 1241 skipped

---

Merge workflow run.

[github-actions]

Test results for "MCP"

5 failed
❌ [chrome] › mcp/cli-webmcp.spec.ts:188 › webmcp-call disambiguates same-name tools in identical same-origin frames @mcp-macos-latest-chrome
❌ [chrome] › mcp/annotate.spec.ts:417 › should cancel browser_annotate when the MCP client disconnects @mcp-windows-latest-chrome
❌ [chrome] › mcp/annotate.spec.ts:446 › should switch screencast to -s session on show --annotate @mcp-windows-latest-chrome
❌ [chromium] › mcp/screenshot.spec.ts:468 › browser_take_screenshot (viewport without snapshot) @mcp-windows-latest-chromium
❌ [firefox] › mcp/annotate.spec.ts:446 › should switch screencast to -s session on show --annotate @mcp-windows-latest-firefox

8700 passed, 1474 skipped

---

Merge workflow run.

[github-actions]

🟢 No failures look caused by this PR

Hi, I'm the Playwright bot and I triaged the failing CI run. All 5 failures are MCP tests that
already fail intermittently on unrelated SHAs, and none of them touch proxy configuration —
the only thing this PR changes.

Details

The diff is confined to launch/context proxy plumbing (browserType.ts normalization,
browser.ts client-cert proxy, fetch.ts proxy agent) plus a new
browsercontext-proxy.spec.ts test. The MCP screencast/annotate/webmcp flows that failed don't
configure a proxy, so there's no path from this change to them. The PR's own new proxy tests
passed.

Pre-existing flake / infra

• [firefox] › mcp/annotate.spec.ts:446 › should switch screencast to -s session on show --annotate — the loudest one: failed 43 of 723 runs (6%) across ~25 different PRs (42856, 42344, 42640, 42391, …) and pushes to main. Clearly bimodal and long-standing.
• [chrome] › mcp/annotate.spec.ts:446 › should switch screencast to -s session on show --annotate — failed 3 of 735 runs, on PRs feat(webkit): roll to r2364 #42748 and fix(chromium): bypass service workers on the storage state page #42664 as well as here.
• [chrome] › mcp/annotate.spec.ts:417 › should cancel browser_annotate when the MCP client disconnects — failed 3 of 735 runs, including PR chore(trace): remove snapshot pointers from action events #42431 and a push to main (fedf7530).
• [chrome] › mcp/cli-webmcp.spec.ts:188 › webmcp-call disambiguates same-name tools in identical same-origin frames — failed 2 of 112 runs, on PR feat(test-runner): support locks in test.describe.configure #42792 and a push to main (07f1a615); never on this PR's branch before.
• [chromium] › mcp/screenshot.spec.ts:468 › browser_take_screenshot (viewport without snapshot) — failed 2 of 736 runs, on PR feat(chromium): expose tab strip metadata via page.tabInfo() #42355 and a push to main; never on this PR's branch before.

All five are on the Windows/macOS MCP bots, which is where the screencast timing tests are
flakiest. I'm a first pass, not the final word — but the history here is unambiguous enough
that a rerun should come back green.

_{Triaged by the Playwright bot - agent run}