Familiarity with a character is a lot like familiarity with a latin root word in English - if you know most or all of the characters that compose a word, you can usually do a pretty good job guessing the meaning of it. Certainly there are exceptions (and there may be more in Chinese than Japanese, which I'm more familiar with, as Chinese uses non-phonetic characters explicitly), but I'd still argue that it's quite useful. In Korean, which has a phonetic alphabet (though based on the same characters), I can always sound out a word, but I'll often have no idea what it means, whereas in Japanese, often I can't read a word out loud, but I know exactly what it means. For every "uptight", there are a dozen 心電図 (心:heart + 電:electric + 図:map = electrocardiogram). It works with spoken language as well, though - often times I'll hear someone say a new word, and I'll ask "is this the right character/syllable breakdown?", and if it is, then I already know what it means.

Edit: To your point regarding English having both connection to meaning and pronunciation - that's definitely true, it's just not as immediately clear/visually identifiable, and not as dense. One could certainly argue that, if native speakers are having serious issues learning pronunciation in a language, then the language makes a poor tradeoff with density there.