What is the advantage of using a different symbol for each word, that offsets the huge disadvantages of having to learn and remember a different symbol for each word?
Especially considering that the spoken language already distinguishes between all possible words through pronunciation (and context in the case of homophones.)
It's hard to explain. In English, spelling, pronunciation, and meaning are all more or less interrelated, right? In Japanese, writing (kanji) correlates to pronunciation and to meaning, but pronunciation and meaning are mostly unrelated to each other. Kanji is what disambiguates them.
So, obviously learning 1000 kanji isn't easy. But doing that is what makes it possible to learn 100,000+ words whose pronunciations and meanings would be otherwise largely unrelated.
It's quite similar to the role that Latin/Greek roots play in English. When you see a word that includes "-graph-" you know it probably involves writing, and similarly when a student of Japanese sees a word with "間 (kan)" they know it involves an interval or space. Throw away the kanji, and your student now just sees "kan" - which means the word will probably involve an interval -- or a barrier, or emotion, or appearance, or a tube, or a building, a warship, a crown, an ending, China, a publication, a government ministry, or.. you get the idea.
A lot of people think that and personally as someone fluent in Japanese (as a second, well rather something like fourth, language) I also sort of feel the same way. However if you look at it without the learned biases, there is a great example where a country with fairly similar language in terms of grammar and sounds that had used to use chinese characters switched to a phonetic alphabet and are not noticeably worse off for it: Korea.
There are way too much homophones and you don't always have the luxury of the context. Learning a symbol for each root (not word!) is not that bad, English spelling is almost as bad, actually.
Spoken language is quite limited compared to written Japanese.
Assuming yes, do their users have significant problems understanding the written text when pronounced in an audiobook? Are there well-known conventions or shortcuts or explanations that audiobook readers insert into their speech to signal the correct meaning of the word?
Do Japanese audiobooks provide evidence for or against the idea that doing away with kanji in writing would not harm understanding significantly?
Fiction audiobooks do exists (although not nearly as common as in English-speaking countries), but audiobooks can't possibly work with non-fiction and especially technical texts unless you are going to use English words for literally every single term. I mean, Japanese has only about 100 moraes and way too much words are just 2-3 moraes long.
Easy to learn, no more trying to guess if it's on/kunyomi, immune to mispronunciation from using a foreign alphabet, the list goes on.