While the International Phonetic Alphabet is absolutely invaluable to the field of linguistics, it probably isn't the most efficient tool for simply learning a foreign language. Physical awareness of the correct mouth positions grounded in the most commonly representative phonics & characters of the target language itself are likely much more practical and effective.
For example, the American English vowels from the IPA chart could generally be represented as:
FRONT: ee / i / e / ae / a
MID: u / o / au
BACK: oo / oe
These could hold down the basic vowel sounds and regular spellings, with alternatives such as "ea / oa / ei" taught as they come up in a common representative word.
(By the way, this particular representation also has the advantage of easily extending into an easy to understand explanation of the Final/Silent-e Rule...)
Of course, at the university level I do think it's fine to introduce the IPA as a useful tool. Some academic abstraction is to be expected there. Also, I think the power of the IPA for learning a second language comes from the layout of the chart itself rather than the universal symbols (which are more useful for linguistics). So I definitely advocate for basic familiarity with the organizing principles behind that chart layout. I just think it's more practical to drop actual phonetic patterns from the target language into the chart than try to memorize all the extra symbols... So I still wouldn't make it the subject of intense study or testing unless there is a strong linguistics component to the course.
Yes, I have personally gotten a lot of mileage out of the IPA in my work in the various fields of linguistics, philosophy, poetics, education, and foreign language study. But I am also exactly the kind of nerd who learned to read and write Tolkien's lovely Elvish script in junior high school. Most students don't have this level of interest in obscurities, and they probably shouldn't be forced to memorize a third-party set of symbols just to learn and use a second language...
I do have a complete method for teaching and learning English pronunciation that ties mouth positions, phonics, and spelling into one comprehensive system that is easy and effective to use because it is grounded in the learner's experience. If you are interested in learning how to teach it or use it, even just the parts that you particularly need, please contact me here.
In the meantime, please enjoy this silly little meme...