Personally, the only people I know who would eliminate kanji from Japanese orthography are either foreigners who can't get them down and a minority of politically extreme (both "left" and "right") Japanese. The use of kanji in literature often exploits not only the homophones but the range of kanji with similar but slightly different meanings to emphasize different nuances in the phrase; another practice is to give kanji the readings for words with readings entirely different from the original ones for puns or again to emphasize certain points. The first pages of the Death Note manga have three examples of these: 人間界 (usually にんげんかい but here given the reading げかい which is usually written 下界), 死神大王 (しにがみだいおう but given the reading ジジイ) and 死神界 (しにがみかい given the reading ここ). All of this can't be duplicated with a kana only orthography or through Romanization.
There are two fundamental errors here IMHO.It's also worth noting that the homophone problem exists because of kanji, not the other way around. If people didn't haphazardly mash characters together, they wouldn't get so many homophones.
1. The use of Sino-Japanese compounds had been inextricably linked with the language since the dawn of literacy in Japan. You can't eliminate homophones just by getting rid of kanji, the words themselves are already an inherent part of the Japanese lexicon.
2. If people didn't haphazardly mash characters together. I personally find this statement rather offensive and ignorant as someone who knows Chinese and Japanese. How can you even say this with a sound knowledge of Chinese characters? Yes, kanji-compounds may seem complex at first glance but they are hardly "haphazardly mashed together", I would say they have more semantic content for the average Japanese (even if he didn't take 漢文 in high school) than Graeco-Latin compounds do for the average Anglophone.
If Sino-Japanese compounds were to be replaced, what would you replace them with? I hope you don't say Graeco-Latin compounds from Western languages; they are seldom intuitive to most westerners and would mean even less to Japanese, it would be like replacing the Graeco-Latin compounds in English with alternatives from Hebrew. Native Japanese compounds could be an option but the same problem exists with replacing Graeco-Latin (including Romance) loanwords in English with Germanic equivalents.
Having said that, I do believe that the problem of homophones has been overstated. Most of the phonetically productive readings have around 1-3 kanji compound variations of which only 1 will be common. The ones with many possible compounds like かんじょう, こうどう also only have a few commonly used compounds. If you mention homophones having a variety of single kanji to represent them, they are hardly used that way, except sometimes to specify what kanji to write.
As for the case of Korean largely eliminating hanja in the mass media, I can't say much, not knowing much Korean but Korean preserves more of the phonetic contrasts from Middle Chinese and early Modern Chinese than Japanese for one, particularly with the final stops and even now, hanja are used to clarify the meanings in case of homophones and are still used in serious academic literature.
An argument made on the basis of literacy is also without basis - Taiwan, Japan and even the PRC have high rates of literacy in spite of the use of the "oh-so-complex" writing system, with it being particularly remarkable in China with the many rural areas.