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> Thus Japanese had settled on having few syllable sounds _before_ there was any way to write them down, so their syllables _can't_ have been limited by their script (at least initially).

You might be underestimating just how huge an influence Chinese has had on Japanese:

https://en.wikipedia.org/wiki/Sino-Xenic_pronunciations#Ling...

Japanese (as well as Korean and Vietnamese) imported Chinese vocabulary en masse, including a gloss of Chinese translation. That made Japanese much more homophonic since the tones got dropped in the process.

Just compare the kunyomi (native) and onyomi (Chinese-derived) readings for some Kanji and you'll see that the onyomi readings are usually single syllables.

"For example, Sino-Japanese words account for about 35% of the words in entertainment magazines (where borrowings from English are common), over half the words in newspapers and 60% of the words in science magazines."

That said, I don't really agree with GP's contention that using Kanji makes Japanese have a simple sound system. Chinese has a somewhere richer sound system even though it invented the characters (especially if you count tones), and Spanish and Italian have very simple sound systems even though they're written with alphabets.