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"1. If the current browsing context is no longer open, return error with error code no such window. 2. Handle any user prompts, and return its value if it is an error. 3. Let cookie be the result of getting a property named "cookie" from the parameters argument." This pattern doesn't work with a proxy implementation since it must read the full request before communicating with the backend that can know things like whether the browsing context is still open. Also the browsing context may close whilst the request is being read. So generally it seems better to delay these checks until after the request is fully processed.
By “proxy implementation” I’m assuming you mean “intermediary node”. You say that they need to read the full request before communicating with the backend. I guess I don’t understand what this means, since intermediaries won’t run any of the remote end steps themselves. They act, as you say, as proxies, or multiplexers.