This is complete and utter nonsense. First of all, plenty of native speakers of, say, Cantonese and other regional languages of Chinese, learn Mandarin using pinyin, and they do just fine.
Second, as a non-native speaker, the advantages of not having to stumble over zhuyin and its problems hugely outweigh any advantage it might be wrongly thought to have. And the benefits continue well past the learning stage.
The /zhun/ example isn't even true. That sound is sometimes broken down for students as multiple sounds, not syllables, for pedagogical purposes when the teachers are trying to get through to a student. That's the case with either pinyin or zhuyin. They explain it as multiple component sounds (not syllables) because the underlying whacky Chinese phonetic theory says it is made up of multiple sounds. (By the way this theory takes about 10 seconds to learn, so the fact that sounds can be broken down into smaller parts is not some mind-boggling obstacle.) If you were taught that it has two syllables, you may want to consider the possibility that your teacher is fuzzy on the meaning of the English word "syllable."
If teachers are using zhuyin, they also break the sounds down to justify or explain how zhuyin works, since zhuyin is so oddly counterintuitive it needs extra explanation to help ease the cognitive burden of using it (which could be the reason it was reformed and mostly dumped decades ago in mainland China to be replaced by pinyin).
Zhuyin does survive in Taiwan, where they keep it around for political reasons. Accepting a superior mainland system would be a linguistic defeat, and would damage their pride, so they cling to an archaic system. They don't want to admit this outright, so they come up with all sorts of reasons to keep it, and they end up not learning pinyin well themselves, so with that handicap they then try to convince naive foreign students that zhuyin is good.
Don't fall for it if you like, let's see... touch typing, reading signs that are written in pinyin, being able to write down the sounds of Characters you haven't learned yet and have them readable by the average Chinese person, and focusing your time and efforts in the right places.
A better argument on your side of the issue would be to say that pinyin is bad because the 'z' and 'h' and 'u' and 'n' are four separate symbols, but zhun is one syllable, so it's (you could say) confusing to the mind (it's not). But that's simply ridiculous. Users of phonetic languages, as all of us are, are perfectly accustomed to seeing multiple sounds and syllables for that matter represented by a number of symbols that does not necessarily correspond to the number of letters.
Mostly the great thing about pinyin is it just does its job transparently without being a distraction. It becomes invisible almost, because since the letters are familiar and the sounds are already readable right off the bat, I can focus on the language and learning words and sounds correctly, instead of having an extra layer in between me and the language.
Pinyin has served me well as a learner, and the results are hard to argue with. I didn't start learning until my 20s, and now pass as a native speaker over the phone. So... don't listen to that guy. Pinyin is great.