If it sounds good to you in English, then yes, it should be accepted. How about "All birds do not sing". Doesn't that mean the same thing?
Technically it doesn't mean the same thing, but what it ends up meaning is so weird that most people would subconsciously change it to make it mean the same thing as "not all birds sing" anyway. "All birds do not sing" is more literally equivalent to "There is not a single bird that sings" or "No bird sings".
Thank you for that answer. Yes, logically, I used to think it meant "No birds sing" but came to realize that people did not usually mean it that way. :)
It's a sentence that is virtually useless because some people understand it one way, some understand it another way, so you can't use it to communicate an idea unambiguously.