Well, yeah they did, but thousands of years before bopomofo script.

Hiragana is at most ~1500 years old.

Many cultures that borrowed Chinese characters developed their own ways of simplifying them. Moreover, Chinese characters themselves are composed of distinct radicals, so it's rather natural to try to decompose them when needed. The radicals are ancient, but they can look very similar to kana in their simple forms. This could be an example of parallel evolution, not one borrowing exclusively from another.

Yes but they still use the radicals in the Chinese characters themselves, so it's odd to say it comes from Katakana/Hiragana.