This 't' is
sometimes realized as a glottal stop [ʔ] - the closure usually made between the tip of the tongue and the dental ridge is made instead by a constriction of the glottis (which is not, in English, involved in distinguishing phonemes).
But I wouldn't question Newbie's 'hard-to-hear' I remember in the late '50s being very confused by Perry Como, in 'Magic Moments', singing 'Time can't erase the memory...'. the context (particularly the phrase 'erase the memory'') makes it clear this is 'can't'; but I had never met the expression at the time. I suppose this just reinforces the point about phonemes; I didn't know the phonemes of Am Eng.
Then why does the language choose this way to indicate a totally opposite meaning?
Ermmm... it doesn't.
Choose that is.
People choose. And they wouldn't have chosen to express themselves in an 'ambiguous' way just for the hell of it. (Which confirms the point that the distinction isn't hard to hear for a native speaker (of American English).
Of course, the problem doesn't arise in Br English (though I'm sure many others do ;-)): /kæn/
versus /kɑ:nt/ - and whatever happens to the /t/ the vowel is still distinct.
b