Can you name one that does not have a phonetic system as well?

Note that Japanese has kana, and Chinese has pinyin and bopomofo/zhuyin.

Other than that... ancient Egyptian hieroglyphics?

Please let me know if you are aware of more.

I'm not the parent, but the ancient sumerian cuneiform script comes to mind.

Egyptian hieroglyphics did have a set of phonetic glyphs as well as the more complex logographic ones

Yes. Egyptian hieroglyphs have a full set of consonants, so you could write it that way. They just didn't because of historical spelling.

As you said Sumerian could be an example of totally "three step cue" system.

Chinese didn't always have pinyin... and Japanese didn't always have kana (it used to be spelled exclusively with Chinese characters, though some characters were used phonetically, and kana resulted from the simplification of these characters).

at least in classic chinese the pronunciation of the script differed by region. several mutually incomprehensible languages shared identical textforms,as far as i understand it.