Early video games didn't have enough resolution to display Kanji, which is why all text was Kana-only. With spaces as word separators and sentence context homonyms are no more of a problem than in other languages.

What do video games have to do with it? Pong didn't have the resolution to display Shakespeare, but we didn't stop using words because of it. Video games provide a visual context that reading alone does not. Also, early video games, especially consoles like the NES Famicom, were geared towards kids who don't understand many kanji. The same is true of books aimed at children in Japan. They don't use kanji and they provide visual context in the form of illustrations.

Ask any Japanese person why they use kanji and not just hiragana, katakana, or romaji and they'll tell you the same thing: Because it is hard to determine the meaning. They don't say it is impossible, just that it is difficult, which it is.

For a contrived example, if I write はしをさがしてる。What am I looking for, a chopstick, a bridge, or the side of something? There is no way to discern from the sentence alone. However, if I use this kanji for hashi, 箸, then it is obvious I'm talking about chopsticks. There is no ambiguity.

This doesn't always help. Sometimes the homonyms use the same kanji. 青山, for instance, can be read あおやま, which is a surname, or it can be read せいざん, which means a lush, green mountain. Most of the time though, kanji provides meaning and clarity where hiragana, katakana, and romaji do not.

This hits you hard when you read children's books for the first time as an adult.

Spoken language doesn't have kanji and people can communicate just fine.

@HumanDrivenDev - Right, because in spoken conversation the listener can ask for clarification when the speaker's commentary is ambiguous, which happens all the time. You cannot do that when you're reading a book.

In mandarin people tend to add lots of little "particles" (not sure about the proper linguistic term) when they speak, which seems to be a way of adapting to homophones.

Spoken language has accent.