I had a quick look at the code. Looks like it is by design that Twitter cards are not added to anything but posts and pages (determined by is_singular() ).
So if your homepage is a page, it should display. If it's a default recent blog posts overview, it shouldn't (at this time).
I suggest someone opens a feature request on GitHub for this to change, if there isn't one there yet.
Would be useful if the feature request included some ideas on what to put in the homepage twitter card, as the title, description, image etc would all be ambiguous or unknown.
Which I presume is why it's not implemented at the moment.
Hope this helps!
Smile,
Juliette
