There aren't any "English pronunciation courses" in the universities that I am familiar with in the US, though as a part of general English as a Second Language they do teach pronunciation. There are also specialized courses in Speech & Hearing science, which generally focus on English. Linguistics departments don't generally have "English pronunciation courses", though they might have a service course that aids students planning to take a transcription course.
Although phonemes cannot be pronounced per se, they can have a phonetic realization, if there is agreement on the mapping from phonemes to surface phonetics. One probable goal of such a course is to teach students of some standing phonetics-to-phonology mapping. There are many actual mappings from the spellings "super" or "how" to a phonemicization (we don't agree on the phonemeicization, sometimes because it matters which dialect you are teaching), and then many mappings from phonemes to pronetic outputs. Getting to the phonemic representation is abstract less useful knowledge in some contexts, if for example the students are not actually native speakers of English and they want to master the pronunciation of some dialect (e.g. Central Valley California English, RP, Capetown dialect).
The problem (which may not be insurmountable) with just teaching narrow transcriptions (for example the dialect of the teacher) is that you can't generalize beyond the examples that you learned about – unless perhaps you covertly develop your own phonemic analysis. That is, if I give you a dictionary of 5,000 English words phonetically transcribed, what have you learned other than 5,000 transcriptions? If I teach you a system where you learn how the phoneme /i:/ is pronounced in various contexts, then at least if you can also figure out what vowel phoneme exists in "bleat", you have learned (in principle) how to pronounce that word.
As for the status of the IPA as a "phonetic" alphabet, while that is what the "P" stands for, the IPA is at heart a phonemic alphabet. Every phoneme must be representable in the IPA, but not every phonetic detail is representable in the IPA (except perhaps in a contrived and non-standard way). For example there are only three length diacritics: long, half-long and extra short, but there are vastly more non-contrastive but systematic degrees of length (especially in Uralic languages). There is no way to write the phonetic quality of "ɪ" in Kikuyu as distinct from the the phonetic quality of "ɪ" in Kamba, you either have to resort to a table of formant values, a description ("a bit lower than in Kamba"), or with recorded samples. Phonetics is about the physical continuum of speech, and no system of discrete letters can represent the ever-changing continuum that exists in an utterance.
At the moment, I can't think of anything that explains the difference between phonetics and phonology in a simple manner. The problem is that we don't all agree on the difference.