Maybe a dumb question but I just want to make sure... Are you using the Share icon (underlined arrow) in the upper nav of an open drawing (make sure you can see all your drawing tools)? That's the one with the option to Send to Illustrator CC.
If that's the icon you're tapping, and still not seeing it, can you send me a screenshot of what you're seeing? (You can send it through the in-app feedback if it's easier.)
It turns out that is not a dumb question. Every instruction I read was pretty vague and just said to press the share button but never showed it or described it. Because this is the icon for share in many, many other applications I assumed this was it. Your description of the share button helped.
In fact it does "share" it by publishing but doesn't have the send to Illustrator option I was looking for.
It's super common on iOS to choose the wrong Share icon because they're both the same, so I was hesitant to ask; at the time it didn't occur to me that the reason it's confusing on Android is because the Share icon is different than what's expected. It makes perfect sense, for the reasons you mentioned, that it's confusing and I'll share your comments with the Draw team.
Glad I asked and glad things are working.