chore: refactor some code around proxies

[dgozman]

Extracted from #35389. No behavior change whatsoever.

[github-actions]

Test results for "tests 1"

6 flaky

⚠️ [chromium-library] › library/chromium/oopif.spec.ts:284:3 › should click @chromium-ubuntu-22.04-node18
⚠️ [firefox-page] › page/page-evaluate.spec.ts:424:3 › should throw for too deep reference chain @firefox-ubuntu-22.04-node18
⚠️ [playwright-test] › ui-mode-test-ct.spec.ts:59:5 › should run component tests after editing test @macos-latest-node18-1
⚠️ [webkit-library] › library/screenshot.spec.ts:44:14 › page screenshot › should work with a mobile viewport @webkit-ubuntu-22.04-node18
⚠️ [webkit-library] › library/screenshot.spec.ts:66:14 › page screenshot › should work with a mobile viewport and fullPage @webkit-ubuntu-22.04-node18
⚠️ [playwright-test] › ui-mode-test-watch.spec.ts:145:5 › should watch all @windows-latest-node18-1

39110 passed, 807 skipped
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Merge workflow run.