Well, I thought I had this all figured out. I went through all the steps, got the Typekit Extension activated, and it correctly shows the Typekit font. But when I select the dropdown in the Font manager, all it shows are the system fonts and Google fonts. I have activated the Typekit Extension, and I have cleared my cache over and over (I have WP Rocket).
I also tried disabling the Typekit font, clearing cache, and then enabling it again.
What am I missing? (Just in case, I'll send you login credentials.)