I'm going to propose that this phenomenon has nothing to do with voiceless stops. I think it's worth pointing out that [ɾ] is an allophone of the voiced /d/ in at least as many accents as [ɾ] is an allophone of /t/. Though, I suspect that /d/ is flapped/tapped in more accents for the reason that Brits so commonly ask why Americans/Canadians/Australians use a D in place of T, which implies they perceive [ɾ] an an allophone of /d/ but not /t/.
In North American English, /d/, the voiced counterpart of /t/, in such
positions is also frequently pronounced as a flap, making pairs of
words like latter and ladder sound similar or identical.
Which might lead one to conclude that flaps/taps are not a product of voicelessness or plosivity at all. /t/ and /d/ are realized as a flap/tap in some contexts because these sounds are all alveolar, while /k/ and /p/ are velar and bilabial respectively. Though there are apparently velar taps and bilabial flaps in existence, these are exceedingly rare, to the point there is not even an IPA symbol for the velar tap, and the bilabial flap only exists (primarily as an allophone of a labiodental flap) in a small group of Central African Languages. This might imply that ease of articulation is a factor in why these sounds are not as widespread as alveolar flaps. (Note: flaps/taps can be voiced or unvoiced.)