> What would a designed writing system look like? I think a properly phonetic/syllable approach like Hangul or even the Kana might be correct.
You're thinking in too broad of terms. A system representing syllables as units would be a bad choice for english, which has extremely intricate syllabic structure (consider "strengths", which is CCCVCCC). This makes a Hangul-like system a stretch, though possible, and a Kana-like system, wherein every possible syllable has its own character, completely impossible. Most[1] potential english syllables are unused or used so rarely that no one could ever be expected to know their character. Douglas Adams named a fictional person "Slartibartfast" -- as far as I know, the syllable "slar" (rhymes with far) has no other existence in English. How would he have written it in an English syllabary?
Chinese has so few possible syllables that enumerating them is quite easy, but it doesn't use them all either. Kana work in Japanese because the only legal syllable structures are CV, V, and N.
> What would a designed writing system look like?
Well, all writing systems are designed; none are naturally occuring. It's hard to know what you mean by this, but:
- A syllabary works fine when the phonology of the language allows for it
- Spanish is a good, though not perfect, example of an alphabetic writing system that corresponds closely to pronunciation (a minor wart would be that I believe 'b' and 'v' are the same sound; c/z in Spanish Spanish and c/s in latin american Spanish have a similar problem)
- The Cherokee syllabary was created from nothingness within living memory
- The design goal of Esperanto was probably similar to what you're thinking of
Anyway, big picture, a given writing system is not equally suitable for all languages, so it doesn't make sense to ask which approach is "correct". They're more and less workable. Alphabets (in which the basic idea is one character per phoneme in the language) are about as simple as it gets, since the number of characters necessary is V + C rather than the O(V * C) needed for a syllabary in the languages with the simplest phonological structure.
[1] I have no numbers for this; it's a guess.