I was in a similar position when I tried to upload from Sketch and Line, but Draw should send to paths immediately.
(Sketch and Line illustrations require one more step, the conversion button, but Draw should just open immediately)
Check the version of your Adobe Draw app? or Illustrator CC? (I'm using 2014.1.1 and it seems to work fine from my iPhone 6)
Do the images from Adobe Sketch send to Illustrator and convert for you?
Illustrator / Libraries
There is a Window option for Libraries.
Additionally, there are ways to access your CC resources from other panels too.
As for the youtube video Working with Adobe Shape & Adobe Illustrator Draw - YouTube
Starting at 5:22, Geoff Dowd supposedly taps the screen, but I don't see him hit the button, do you?
I did - worked just fine (but I regularly update my CC apps and iPhone/iPad apps)
hope this helps!
It's working now! My Creative Cloud was showing that all my apps were up to date but when I opened up the Update Management Tool I was able to see a new major update that included the CC Libraries I wasn't getting access to before. I should have checked that earlier. Now my Illustrator CC is updated to 2014.1.1 and everything is working as it should.
Thank you so much for the help! I really appreciate it