Proponents of phonetic spelling tend to be more pretentious than the "pedants" they're lambasting. There are any number of problems with phonetic spelling, but I'm only going to go over the ones which are to me the most obvious.
First of all, with traditional locked orthography, it is possible to read much faster. There's no two ways about it: you can read something spelled correctly faster. This is because the human eye will begin looking for the shape of words, instead of going over all the individual letters. This is also why really unusual fonts are more difficult to read.
Notwithstanding, today, English is spoken on several continents. Were we all to spell everything phonetically, it would be easy to render our prose impossible to read by people not from our particular region. For example, some people might pronounce "can" as it's spelled, while others might say something resembling "ken." Reading text littered with strange pronunciations would be a leviathan task, as when reading it is not as easy to compensate for dialects as it is when listening. This will only get worse in the future, as the English language evolves and its pronunciation changes.
Some argue that because authors like Shakespeare didn't write in a standardized form of spelling, we shouldn't either. That's absurd. Firstly, in Shakespeare's time, everyone for whom he was writing lived in England, a fairly small place. Today native English speakers are found on either side of the Atlantic Ocean. Furthermore, you'd be hard-pressed to find totally outlandish spellings like *"u" instead of "you" or *"uv" in place of "of" even in the works of Shakespeare. The spellings for the most part resemble the spellings you'd find in a modern dictionary.
The Latin alphabet, in which most Western European languages are written, simply doesn't lend itself to phonetic spellings, especially the way it's interpreted in English. There are many different ways to get the same sound, different sounds for the same letters, and combinations that totally change the sound of both letters. It's simply not another hiragana or hangul. Were we to use phonetic spellings, it would also mean dictionaries would have to be re-organized, and consistent use would have to be delineated for the dicitonary to be at all useful. Of course, it is likely that dictionaries wouldn't do this the same way.
In conclusion, you'd be better served spending all the energy you spend kvetching about how the traditional orthographical system is unfair into learning to spell.