In addition to the phoneme problem, it's about readability. Yes, really. The first time I saw わたし written as 私 I just about instantly remembered the latter (it is, after all, used constantly in writing). That kanji is much easier and faster to read than the corresponding hiragana, and it was like that from way back when I had just started learning Japanese. I still have a way to go.. learning a language at my age turns out to be quite slower than when I was younger.. but everything is just easier to read, as soon as one's able to read something in kanji instead of hiragana. The latter is hard and slow to read, even though it's such a simple character system to learn.