chore: tidy up headless-shell hacks

[mxschmitt]

This addresses #33964 (comment).

[github-actions]

Test results for "tests 1"

6 flaky

⚠️ [chromium-library] › library/popup.spec.ts:264:3 › should not throw when click closes popup @ubuntu-20.04-chromium-tip-of-tree
⚠️ [webkit-library] › library/browsercontext-clearcookies.spec.ts:92:3 › should remove cookies by domain @webkit-ubuntu-22.04-node18
⚠️ [webkit-library] › library/browsercontext-reuse.spec.ts:86:1 › should reset serviceworker that hangs in importScripts @webkit-ubuntu-22.04-node18
⚠️ [webkit-library] › library/browsertype-connect.spec.ts:368:5 › launchServer › should reject waitForSelector when browser closes @webkit-ubuntu-22.04-node18
⚠️ [webkit-library] › library/proxy.spec.ts:178:3 › should exclude patterns @webkit-ubuntu-22.04-node18
⚠️ [webkit-page] › page/page-set-input-files.spec.ts:205:3 › should upload multiple large files @webkit-ubuntu-22.04-node18

37247 passed, 650 skipped
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Merge workflow run.