The story of how Sequoyah created this is a good one. He knew of the technology of writing and saw its power but essentially did a clean-room re-implementation of it.
The first incarnation was closer to Chinese or ancient Egyptian in that he tried to create a different symbol for each word. Then, after creating over a thousand symbols and still being far from his goal, he went back to the drawing board and took a purely phonetic approach.
Not being literate in English, he simply borrowed sritten symbols he saw, including numerals. Ultimately, the phonetic-based system was a success and spread through the Cherokee nation quickly.
Humans have only invented writing independently two or maybe three times. It's amazing how much easier it becomes just knowing such a thing is possible!
What is also quite fascinating to me is that many of the Cherokee of the time considered writing to be a form of witchcraft. Sequoyah allegedly [e.g., 1] decided that it was not based on his observations of white men, but his initial trials of teaching written Cherokee were sufficient to bring charges of witchcraft against his daughter, his first pupil. However, after some demonstrations, it was shown to be what it is, a very simple but extremely effective tool.
I'm sure many of the readers of this fine forum know the quote (and provenance, which I don't) that is something like 'any sufficiently advanced form of technology is indistinguishable from magic'.
The evidence we have from Egyptian, Mayan, Sumerian, and Chinese [1] shows that the evolution of writing tends to follow a similar pattern. You start with (simplified) pictorial representations of words. Extend this to words that are difficult to draw by use of homonyms or other phonological puns (the rebus principle). Eventually, you simplify the system by using a phonetic principle, largely based on syllables [2]. Sequoyah did follow these steps, but he used pre-existing knowledge of what written English looked like to arrive at the end in rather less time than ancient civilizations took.
An interesting note is that somewhere in the development of Egyptian writing, the vowels got dropped from the syllable representation, leaving an abjad which the Phoenicians borrowed... and then the Greeks added vowels to obtain the alphabet, while Sanskrit took a different path to get the abugida. As near as we can tell, this one turn away from a syllable-based phonetic system is unique in the history of independent development of writing systems.
[1] Indus Valley Script is undeciphered, and may not even be a full writing system, and so it is excluded here.
[2] This did not happen for the Chinese, but the Japanese borrowed the Chinese script and made two syllabaries out of it, hiragana and katakana, following the same basic principle.
The greeks adopted the phoenician alphabet, even though phonologically phoenician and greek languages have a lot of differences. I have read of a theory claiming that greeks simply misheard the sounds of the phoenicians and made the best adaptation for greek. So for example, alef [1] became a vowel in the greek alphabet, since the glottal consonant was not present in greek.
> ancient Egyptian in that he tried to create a different symbol for each word
This is not how Egyptian hieroglyphs worked.
An common example
𓉐𓂋𓂻 - pr (v) - to leave
𓉐 is a house. It represents, here, the sound pr, which is the Egyptian word for house.
𓂋 is a mouth. It represents the r sound. Here, it's being used as a phonetic complement to emphasize that we're spelling a word here and not talking about a house.
𓂻 is a pair of legs. It's being used as a determinative to tell you the word has something to do with walking.
Related (as they were influenced by the Cherokee writing system) and similarly interesting: Canadian aboriginal syllabics [1], which are unique in that the shape of the character determines the consonant, while the 90-degree rotation of the character determines the vowel.
Strictly speaking, the Canadian Aboriginal syllabics is an abugida, not a syllabary.
In the categorization of writing systems, alphabets (such as Latin) encode each phoneme as a distinct letter; abjads (such as Arabic) encode only consonants, with vowels as optional diacritics; abugidas (such as Indic scripts) primarily encode consonants but mark vowels in some systematic way; syllabaries just encode each syllable as a unique glyph without any systematic modification; and logographic systems don't have any systematic phonetic component (although they usually are influenced by phonetics, e.g., the rebus principle of diagramming abstract concepts by use of homonyms).
Sequoyah had access to an English-language Bible, so although he couldn't read the Latin text, he used several letters as inspiration for Cherokee, which is why you see Latin characters that correspond to completely different sounds.
While I agree the distinction you reference between abugidas and syllabaries is useful, it's not a distinction that's always observed (e.g. Indic scripts are sometimes referred to as syllabaries).
I have family who are Vai, a tribe from Liberia/Sierra Leone. The Vai language has a syllabary, one of the few independent written languages in Africa. As it says in the article, there is a possible connection with the Cherokee syllabary. When free African-Americans got the option to settle in Liberia some Cherokee opted to join, apparently not seeing a future in their colonized homeland.
For people who might be interested in the modern Cherokee (including lessons on the language, food and culture) might enjoy a Cherokee produced show called Osiyo http://osiyo.tv/
The first incarnation was closer to Chinese or ancient Egyptian in that he tried to create a different symbol for each word. Then, after creating over a thousand symbols and still being far from his goal, he went back to the drawing board and took a purely phonetic approach.
Not being literate in English, he simply borrowed sritten symbols he saw, including numerals. Ultimately, the phonetic-based system was a success and spread through the Cherokee nation quickly.
Humans have only invented writing independently two or maybe three times. It's amazing how much easier it becomes just knowing such a thing is possible!