We seem to be discussing at least three separate issues in this thread:
1. Is gotta a modal verb?
2. Should we teach it?
3. What about gonna, wanna, woulda, shoulda, coulda, would of, should of, could of?
The answer to  is simple: NO. By the criteria most writers today accept for identifying modals, there are just nine core modals: can, could, may, might, must, shall, should, will and would. Among the generally accepted criteria, just two will suffice to reject gotta:
in forming negatives, the modal is placed before the negative word not –
√ he will not do it; ╳ he gotta not do it
in forming interrogatives, modals use inversion:
√ will he do it?; ╳ gotta he do it?
The answer to  surely depends on the students. I agree with Heterological when he writes, “I'm all in favor of teaching reductions and contractions to students. They're not at all the same as "slang;" they're important features of native speaker pronunciation.” However, for learners who need to write formal English and/or pass examinations, I would make it clear that, except in the most informal writing, these are spoken, not written forms.
Part of the answer to  is the same as for . They are normal (not ‘non-standard’, Raymott) pronunciations in the informal speech of very many native speakers, including educated ones. To this I would add the fact that few educated speakers would write gonna, wanna, and I suspect none would write would of, should of, could of, (because, as Heterological explained, they are incorrect). But it’s the spelling that is incorrect, not the pronunciation!
My final points are in response to Raymott’s questions:
Do British English teachers teach Scottish variations like 'canna'?
No ,because canna is a dialect word, and I would not teach it except on a university course specifically looking at dialects. However, the expressions we have been discussing, better written as (for speakers from southern England) /gɒtə, wɒnə, wʊdə(v), ʃʊdə(v), kʊdə(v)/, are not dialect expressions; they are normal conversational spoken forms.
and do they cover all regional accents with special phonetic spellings?
On a university course specifically looking at dialects, yes – if by ‘phonetic spellings’ you mean using IPA symbols. On standard English courses at elementary levels I would not spend time on the dialects of English. However, on more advanced courses, I would draw the attention of students to some of the more important differences in pronunciation (and in vocabulary, grammar and spelling) between American and British English.