[dartpad_ui] Use the new cross origin window support from pkg:web

[parlough]

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[github-actions]

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[devoncarew]

For my own edification - this is nullable? Or, won't throw in cross origin situations?

[parlough]

window.parent was nullable as well, this just won't throw when calling its members (postMessage) when cross origin due to implicit null or types assertions that the web compilers would add.