Oh, quite, but there were still HUGE cultural ties between China and Japan.
Why the Philippine would adopt the Korean variant of the Chinese writing system, I'm sure I don't know.
But a stronger Chinese maritime presence (trading, earlier colonization of Taiwan, etc.), could, IMO, easily lead to a Philippines that was Sinicized culturally in a similar way to Korea and Japan, although not as thoroughly, I'd imagine.
OTOH. Once you get away from Sinitic languages, the Kanji/Hanzi/Chinese character system is inadequate to express the language. Which is why Japanese developed kana (hiragana/katakana) to express the grammatical and other bits of Japanese that Kanji doesn't work on.
If your various Philippine languages did the same, the 'glue' bits (kana-equivalent) would be different from one language to the next (although, as pointed out similar), so they'd lose the 'one writing system for multiple languages' advantage that worked so well with the various Chinese languages.