fix: chrome.action API registration

[codebytere]

Description of Change

Closes #36545

Fixes an issue with correct uncompiled source registration:

[12932:1013/112709.120:ERROR:extensions_browser_client.cc(69)] Extension Error:
  OTR:     false
  Level:   2
  Source:
  Message: Error handling response: TypeError: Cannot read properties of undefined (reading 'setIcon')
    at setExtensionIconAndPopup (chrome-extension://fmkadmapgofadopljbjfkapdkoienihi/build/background.js:1:1614)
    at checkAndHandleRestrictedPageIfSo (chrome-extension://fmkadmapgofadopljbjfkapdkoienihi/build/background.js:1:2090)
    at Array.forEach (<anonymous>)
    at chrome-extension://fmkadmapgofadopljbjfkapdkoienihi/build/background.js:1:2146
  ID:      fmkadmapgofadopljbjfkapdkoienihi
  Type:    RuntimeError
  Context:
  Stack Trace:
    {
      Line:     0
      Column:   1
      URL:
      Function: (anonymous function)
    }

This was happening because action.json is a different type of extension impl format, and we need to register functions from the sources. This fixes the issue, which means that React DevTools finally loads without errors 🥳

Also adds some tests to prevent this regressing.

Checklist

• PR description included and stakeholders cc'd
• npm test passes
• tests are changed or added
• PR release notes describe the change in a way relevant to app developers, and are capitalized, punctuated, and past tense.

Release Notes

Notes: Fixed a unexpectedly thrown error in some unsupported chrome extensions.

[release-clerk]

Release Notes Persisted

Fixed a unexpectedly thrown error in some unsupported chrome extensions.

[trop]

I have automatically backported this PR to "26-x-y", please check out #40512

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I have automatically backported this PR to "27-x-y", please check out #40513

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I have automatically backported this PR to "28-x-y", please check out #40514