Well the question was what do they have that comes closest. They do have a lot more than just pinyin and handwriting. If you don't know, it's fine to just say that, or stay away from the discussion, especially when you seem to be answering with such certainty on something where you didn't read the question.
They have nothing that resembles the tenkey input because they have nothing that resembles kana, unless you want to count bopomofo. If you want to make some correction, rather than just condescending to me, feel free.
Sorry I just find statements of the form “nothing exists”
a bit irksome especially when the person talking doesn’t know what exists as can be seen by looking at the matter at hand. Chinese has plenty of ten key input systems, some alive, some dead, since the mid 1990s at least for phones and possibly before that for the keypads accompanying computers. It’s been a very fertile space for innovation and I’d be extremely surprised to meet anyone who has kept up with it, so forgive me for being skeptical of your claim of perfect knowledge of a negative. I was asking a different person who sounded like he knew something, and you answered. I did mention zhuyin which is commonly understood informal shorthand for zhuyinfuhao, the more formal word for bopomofo. There are component based systems as well that work with radicals, strokes, quadrants of the characters, and all kinds of zany stuff. That being said, I am not very familiar with Japanese and didn’t know about the the flick method.