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.