That's what's shown for the phonetics, yes, but we aren't talking about phonemes, we're talking about phonological effects, specifically the anticipatory assimi

That's what's shown for the phonetics, yes, but we aren't talking about phonemes, we're talking about phonological effects, specifically the anticipatory assimilation of [n] into the preceding vowel. (Note: One wonderfully weird thing about English is that different types of phonological assimilation happen in different directions, e.g., voice assimilation happens the other way, so "fads" is pronounced as if it had a "z" at the end and "putz" is pronounced as if it had an "s" at the end, i.e., "fadz"/"puts" rather than "fats"/"pudz")