True on the smartphone. On the computer most Japanese speakers I know just type romaji. However, this is pretty much irrelevant to this article, as romaji->kanas (the phonetic alphabets) is a pretty straightforward and solved problem (there is a clear bijection between both).
The real problem is transforming the phonetic transliteration into the correct word in either kanji (for most Japanese words) or katakana (for words with foreign origin).
This problem is akin to disambiguating between two homonymes (which are much more frequent in Japanese). In some cases it is easy by looking the previous words, but in some it is heavily context dependent.
Nowadays, most japanese typing system will propose a list of kanjis as you type that corresponds to the most frequent writting of your transliteration, but sometimes for unusual kanjis o(or people's name) you have to dig deep into the list.
I can see how such a system could improve typing speed in Japanese.
Do people really use romaji on keyboards in Japan? This strikes me as an odd way to type, as it means that you first need to learn romaji in order to type. I thought that hiragana keyboards (hiragana mapped onto the normal layout) were the norm, especially on laptop keyboards.
But none of the Japanese I know use the hiragana directly. They told me it is mostly old people who use it. Almost every Japanese knows romajis now, there is no additional cost of learning a new alphabet.
Not sure what you mean by dial type input system, but if you mean that people use the number keys 0-9 to input text, that's the exact thing the grantparent meant.
The real problem is transforming the phonetic transliteration into the correct word in either kanji (for most Japanese words) or katakana (for words with foreign origin).
This problem is akin to disambiguating between two homonymes (which are much more frequent in Japanese). In some cases it is easy by looking the previous words, but in some it is heavily context dependent.
Nowadays, most japanese typing system will propose a list of kanjis as you type that corresponds to the most frequent writting of your transliteration, but sometimes for unusual kanjis o(or people's name) you have to dig deep into the list.
I can see how such a system could improve typing speed in Japanese.