"Phonetic" in this context is relative term.
Our script is somewhat phonetic: compared to the logograms of Chinese writing, one might describe our latin alphabet as strongly phonetic - though even then the difference is not as stark as you might assume. Compared to Japanese kana, perhaps our alphabet seems a little less purely phonetic and a little more reliant on remembering contextual rules - a little more mired with special cases and exceptions that arise from certain sequences of letters but only in certain situations.
Compared to the International Phonetic Alphabet, which is based on the latin alphabet but modified to be a truly unambiguous phonetic notation, the pronunciation rules for our writing system seem like a tangled mess of culture and history that must be learned to have any hope of really understanding when a given letter is voiced one way, and when it is voiced another. And even IPA requires extensions to capture aspects of pronunciation that were not as often considered when the system was originally devised.
In short: it's complicated, and what we've got works pretty well. There's still plenty of room for experimentation. If you try a different approach to teaching the alphabet to your kids, I'm certain that there will be phoneticians who would be very interested to hear how it goes! :)