When I was hired by an accent modification company based in New York, I was specifically told not to use the term "phoneme," because it was considered too technical. A phoneme is just a term for the smallest unit of speech in a language. Each phoneme has a specific symbol that represents that sound in the International Phonetic Alphabet. Any respectable accent reduction coach will not only know the International Phonetic Alphabet (IPA), at least as it pertains to the English language, but will also end up teaching the phonemes of English anyway. There's really no way around it because, let's face it, written English is a mess.
There are some rules for knowing how to pronounce a given word in English, but they are few and the ones that do exist have many exceptions. English also doesn't use accents or any diacritic symbols to help the reader pronounce words easily. All of this leads to the necessity of using phonetics to make English pronunciation clearer to the learner. And, by the way, the same phonetic symbols serve for any language - not just English.
Of course, native speakers don't require learning phonetic symbols to learn English. But since it's their first language, there's no risk of confusing the sounds with another language. When learning a foreign language, we are often taught using the written word. While it is not necessary, nor perhaps even recommended to learn English this way, it is nevertheless important to most people to know how to read English as well as speak it.
If you peruse any university foreign language program, a course teaching the phonetics of the language will most certainly be listed as part of the curriculum. When I was working on my BA in Spanish, I had an entire semester-long course in Spanish phonetics - and Spanish is largely a phonetically-written language anyway! That class, however, was extremely important to my understanding of Spanish pronunciation and I'm grateful that it was required for the major. When I later moved on to learning Portuguese, the first thing I did was to buy a dictionary and study the page on Brazilian Portuguese phonetics. This gave me a jump start on learning the language and greatly reduced the transition time from Spanish to Portuguese.
My last point regarding this topic of learning English phonetics is this: How hard is it really to learn a dozen or so phonetic symbols if it means clearing up many questions about English pronunciation? After all, you have already come this far with your English skills. Additionally, you've probably studied a great deal to be good at your profession. That was likely much more difficult.
So join me for a 12-week accent modification course and I'll make the experience as painless as possible. Who knows? You may even find learn English phonetics enjoyable.