Growing up, I always wondered why the abbreviation for eccetera/eccetra (/ekˈsɛ.ɾ(ə).rə/ or /ekˈsɛ.trə/) was “etc.” and not “ect.” Finally, I looked up why in college and found out that “etc.” actually stands for et cetera. Since then, I’ve pronounced it /et ˈsɛː.ɾ(ə).rə/ because this pronunciation matches what “etc.” actually stands for.
However, as other answers have pointed out, originally et cetera was pronounced /et ˈkeː.te.ra/ in Classical Latin and /et ˈt͡ʃe.te.ra/ in Ecclesiastical Latin, so my pronunciation still isn’t “correct,” but at least it makes me happy because it matches the order of etc. and is an American English phonetic reading of et cetera.
That being said, et cetera became eccètera in Italian, so an t to c shift in et cetera is common in other languages, not just American English. This gives some weight to those who want to go for the /ek/ pronunciations of et cetera.
Argument for /et ˈsɛː.ɾ(ə).rə/: This pronunciation not only is an American English phonetic reading of what “etc.” stands for, which is et cetera, but it also matches the letter order of the abbreviation “etc.” This helps me fall asleep at night.
Argument for /ekˈsɛ.t(ə).rə/: The same phonetic process that turned et cetera into eccètera in Italian probably made created the /ekˈsɛ.t(ə).rə/ pronunciation in American English, so let’s embrace our “cultured” Italian pronunciation of et cetera! Who cares if “etc.” doesn’t match the letter order of what we’re saying? Et cetera never was pronounced /et ˈkeː.te.ra/ anyway, and I don’t think anyone is going to start saying Classical Latin “et-ketera” anytime soon.
Argument for /et ˈsɛ.trə/: It follows the order of the abbreviation “etc.” and is a loose phonetic reading of et cetera, and it sounds less academic than /et ˈsɛː.ɾə.rə/.
Argument for /ekˈsɛ.trə/: I’d say the same arguments hold true as for /ekˈsɛ.t(ə).rə/. It’s just whatever your personal preference is.