>The writing system that he invented was a syllabary, not an alphabet like the people he'd been observing or a logographic system like some other Native American languages.
For some reason, the alphabet seems to be extremely difficult to invent. IIRC it only happened once, with the Jews laying down the foundations and then the Greeks actually taking it all the way with vowels (I shudder to think what the world would look like if the Bronze Age collapse hadn't wiped out literacy in Greece). OTOH syllabaries/logographic systems have been invented independently a _lot_ of times all over the world.
Writing has developed independently between two and five times (Mesoamerica and Sumer for sure, maybe China, Egypt, India in decreasing order of likeliness). All of these start with a logographic script, which then gets filtered into a syllabalic component. Egypt dropped the vowels from its representation, having its characters represent only consonants--this was adapted by the Phoenicians. The Greeks took the Phoenician script, repurposed some of the characters representing consonants they didn't have into vowels, and created the first true alphabet (whence Latin, Cyrillic, Georgian, and Armenian, basically every true alphabet in the world). The Arabics and Hebrews adapted the script into their modern abjads (where vowels are marked, if ever, by diacritics), while the abugidas developed (where vowels are indicated by systematic manipulation of consonants) in India by debatable means.
Alphabets, abjads, and abugidas were effectively independently invented only once (it's debatable to what degree the creators of Hangul were aware of other alphabets). Syllabries appear to be the very natural development of writing.
It was the Phoenicians who invented alphabetic writing (well, actually an abjad), not the Israelites, who, as with the Greeks, borrowed and adapted the Phoenician script.
For some reason, the alphabet seems to be extremely difficult to invent.
You might be right, although the one known invention of it spread around pretty well, seeing as the Phoenicians primarily got around by sea. And I'm not sure it's always needed: while I'm not familiar with Cherokee, to the extent I understand Japanese, at the lowest levels it's encoded as a syllabary because that's the way their language works, and the ideographs they borrowed from China known as kanji in part help to distinguish between the many homonyms that are inherent in such a language, with the relatively few exceptions to strict use of syllables either easily encoded or a matter of spoken dialect.
Japanese was first written in kanji (borrowed Chinese characters), with the syllabaries created hundreds of years later. Though you could say modern Japanese is encoded as a syllabary.
Do you happen to know how japanese input methods generally work?
I have seen chinese and japanese steno machines before, e.g. chorded strokes that cover one or more sillables. That takes specific hardware and a bunch of training, though, so I'd guess it isn't the primary one.
I'm a beginning Japanese learner. I really know nothing about the language. But here is how it might be typed (at least by a non-expert like me):
Suppose you're sitting at a keyboard with Roman letters. If you want to type a sentence, you might first type the "anofuyuhataihensamui" keys. The IME would convert that into a stream of Hiragana characters as you type to get あのふゆはたいへんさむい.
To do that with a cell phone, you would have to press keypad keys to get the character you want. This isn't too bad since there are only 46 basic Hiragana characters. iOS and android have nice "flick" interfaces for this: https://www.youtube.com/watch?v=2LY0-8-PXSc#t=125s These interfaces are efficient because they're adapted to the small set of Hiragana characters you write. Some teenagers can (painstakingly) write entire novels on the cell phone like that.
Hiragana characters can express all the sounds of the Japanese language. Unfortunately, they're not very readable because there are no visible word breaks and because Japanese has lots of homonyms. To make the above readable, you need to make a second pass over the sentence you wrote, converting ふゆ into 冬 and たいへん into 大変 and さむい into 寒い with the arrow keys and space bar. I think it's common to do this after every word so you don't get too far behind. At the end, you get the sentence you want: あの冬は大変寒い, after a dozen or so keystrokes of overhead.
Some modern IMEs try their best to convert kana to kanji as you type. Apple's Japanese IME, for example, converts that sentence without any extra input. But frequently you have to do at least a little bit of post-processing.
I think it's common for actual Japanese people to type using an American keyboard by inputting the romanization of the words they use. There are keyboards that have one key per hiragana character, but I think they're not as common.
From what I understand, Chinese is frequently typed with Pinyin, another romanization scheme. There are more challenges though: first, the lack of a syllabry like Hiragana means there's no intermediate representation. Second, Mandarin uses tones on each syllable which aren't captured in Latin letters. This can also be different between different dialects and geographic locations. Taiwan, for example, has some interesting differences compared to mainland China: https://www.quora.com/Do-Taiwanese-use-Pinyin
For some reason, the alphabet seems to be extremely difficult to invent. IIRC it only happened once, with the Jews laying down the foundations and then the Greeks actually taking it all the way with vowels (I shudder to think what the world would look like if the Bronze Age collapse hadn't wiped out literacy in Greece). OTOH syllabaries/logographic systems have been invented independently a _lot_ of times all over the world.