Is that cellphone argument even really an argument? Apparently in the West most write about 70 characters per minute per hand. So frankly if you have paper ready the speed difference for such short stuff like what you'd want to order etc. compared to typing in your smartphone is probably negligible; in Japanese that's a few kanji or whatever and you are done. Plus, text size is obviously much larger the way she goes about it ...
School might also have a ban against smartphones (unless they were in a scene somewhere, I didn't check). Either way I wouldn't be surprised if some genuinely preferred this method.
I may be just getting old, but phones are primarily associated with calling people for me, so a mute person having one didn't even come to mind as a possibility. Might be the same for her and/or her parents as well, so even if it actually does have useful features, they'd just dismiss it as a silly idea without even realizing there's more to it.
Of course, tablets might still be a similar good idea with a more readable screen for "talking" with people. Pen and paper would still be quite a bit easier to handle, though, and you would not have to worry about charging them.
