Indeed, I would go as far as to say that not glottalizing the /t/ in such words (preceding unstressed syllabic /n/) could sound somewhat odd or affected in Amer

Indeed, I would go as far as to say that not glottalizing the /t/ in such words (preceding unstressed syllabic /n/) could sound somewhat odd or affected in American English. "Important" is an interesting word in American English. I think for many speakers including me the first /t/ is glottalized (following the general rule noted above) but for me at least there's a sound preceding the glottal sound which I don't know how to describe other than "t realization sound". I feel like I've noticed younger (than me) American speakers omitting that first sound and just using the second (normal /n/-preceding) glottal sound. I think I've only noticed this in the Boston area where I currently live but it may be more widespread. To me it makes the pronunciation of "important" sound slightly more like how I'd expect a stereotypical southern England pronunciation to sound, except for the /r/. I have also heard Americans pronounce "important" with the /t/ realized as (I think) a flap and no glottalization, which sounds sort of affected to me, as though the speaker is trying to avoid the glottal sound.