CELTA phonology

On page 156 of The CELTA Course Trainee Book (Cambridge, 2007) by Scott Thornbury and Peter Watkins, the authors discuss the vowel and consonant sounds of English and remark that "[f]or teaching purposes, the sounds of English are often displayed in the form of a phonemic chart":

In the phonemic chart on the left are listed the vowels and consonants of RP, whereas in the one on the right you can see the symbols corresponding to the sounds of North American English. (It should be noted though that on page 155 the authors refer to RP using the wrong expression "Standard British English", thus confusing an accent of English with a dialect.)
As is evident from the image above, the systems of transcription used in the charts are similar but not the same. The one on the left in fact conforms to the principles of the International Phonetic Association (IPA); the one on the right doesn't, as it follows a different AmE tradition, namely that of Trager and Smith. This tradition analyses the English long vowels and diphthongs as consisting of a short vowel plus one of the semivowels (called "peak satellites") j, w, h, the latter being realised postvocalically either as a schwa offglide or as the lengthening of a mid or low vowel. In addition, stress is marked by putting acute or grave accents over vowels (é, è), not by putting marks high up before the stressed syllable as in the IPA.
For me both systems of transcription are perfectly ok; it's just that I don't think it's necessary having to teach or learn two relatively different systems of notation at the same time. Indeed, it's a complete waste of time! Also, it seems as though Thornbury and Watkins were suggesting that you can only use IPA for RP and the Trager-Smith system for General American, which just isn't true. Either system will do! Wouldn't it be much easier then for both teachers and students alike to go for just one system without having to resort to using two systems in order to highlight the differences in vowel realisation and inventory between RP and General American? Phonetics and phonology is already a very complex part of the teaching of English: why do you want to make it more complicated than it is?
By the way, if I had to choose between one system or the other, I would definitely go for the IPA one. The Trager and Smith tradition in fact isn't trendy any longer and it wasn't so even when John Wells in the 1980s was writing his Accents of English.
Why then teach the Trager and Smith notation system when almost all the pronunciation dictionaries and EFL textbooks on the market nowadays make use of the IPA script?