It doesn't work due to the insane amount of homophones. When you are speaking with someone you have context and you can discern the meaning of what it's said. But random words or texts can change it's meaning depending of what character is used. And the tone system doesn't help as it can be seen in Chinese pinyin.
Why do you have less context in writing than in speech?
I'd be willing to bet heavily that the vast majority of those "homophones" are primarily writing-only, ___domain specific or archaic "shorthands", which are referred to in speech with slightly more verbose alternatives. Switching to a non-character based system would admittedly in that case mean some ___domain specific writing would be slightly less compact, but that seems a reasonable tradeoff given the unwieldiness of the current writing system.
You'd lose your bet. In that "shuu" link as an example, most (10-12 or so) are common enough that you might hear them in a typical newscast, with that pronunciation.
What makes things manageable is the combinatorics. E.g. there are dozens of kanji read "shuu", and many dozens more read "kan", but most of them are only read that way when part of a 2-character compound, and only a small subset of the possible "shuukan"s are words, and only a subset of those words are common in spoken conversation.
Even then, it is a very homophone-heavy language. I can think of four "shuukan"s off the top of my head that you might hear from a newsreader; it would only be after those that you'd get into ___domain-specific words. This is pretty typical.
It's not that you have less context, as much as you /need/ less context. Instead of a few extra words to describe something, you get a different character.
In the above example all three are read as おば (pronounced: oba). When spoken you still need to differentiate, but it'd either be obvious from context or you'd just explain it manually.
Because you are going to select different phrases and words while speaking than you are when writing. Even with context clues from the conversation, it can at times be confusing, so you have to explain what you meant. Usually it's verbal, sometimes it's 空書 (sky writing). To avoid having to do this frequently, people will often adopt a subset of the language that is less prone to confusing homophones for their vocal communication.
It creates a situation where you have people who have wildly different voices in their writing than they do in their everyday speaking, which is an interesting phenomenon. (To me, at least)
Korean has just as many homophones and used to be written with characters like Japanese. Now they are doing just fine with their phonetic writing system.
Korean didn't work at all under the imported Chinese system. Japanese also had problems but they solved inventing the Hiragana, a syllabary system. Both languages chose different systems and both languages work fine.
Also Korean avoids many homophones thanks to it's 10 vowels. Japanese has 5.
In addition, Korean spelling is heavily morphophonemic, which is a fancy way of saying that words are written based on its "base form" even when the actual sound is different due to interaction with grammatical suffixes.
A bit like English "packed" being written with "-ed" even if it sounds identical to "pact". Helps disambiguation.
(Actually, come to think of it, it's rather analogous to the Japanese way of maintaining the same Kanji while the suffix changes.)
Koreans did have an old writing system[1] made of Chinese characters, where some were used for meaning and others were used to denote Korean suffixes with a similar sound (kinda like how Hiragana started out, I guess). But it eventually died out.
There's a tongue twister in Chinese with all characters pronounced "shi".[1][2] Hmm, I don't know Japanese grammar, but if those characters are Chinese ones, you can almost write something like that with these characters. 秋收 and 修習 are legitimate words for starter.
All characters are pronounced "shi" in Mandarin, with 4 different tones, leaving 4 distinct pronunciations.
In other "dialects", such as Cantonese or Teochew, the characters are pronounced as 7 or distinct syllables, with 6 different tones, leaving more than 20 distinct pronunciations.
Mandarin has very few available syllables compared to other languages (not only, say, English, but also older Chinese "dialects").
Homophones aren't a problem. Just use a silent radical in front of kana sequences. Swapping 2000 kanji for about 200 radicals is a good enough "90% solution". Some possible examples from the first page of results from you link:
For example, how many kanji can be read as 'shuu'/しゅう: http://jisho.org/search/%E3%81%97%E3%82%85%E3%81%86%20%23kan...
Try to do that with tones.