Choosing aspirated unvoiced consonants as "typical" of English is very weird. Native English speakers do not consciously distinguish aspiration but they do distinguish voicedness so it seems it would be more accurate to call the aspirated form the allophone (aspiration only occurs at the beginning of words and stressed syllables, and never after "s" as you point out, so the unaspirated form probably occurs more frequently too).
Still, I agree with your main point. The Latin alphabet is not used like the IPA by most languages, even English has not preserved all the original Latin sounds as used by the Romans. And there's even precedent for some of the sound choices made in Pinyin that would seem completely alien to English speakers. For example "c" is used in all the Slavic Latin alphabets (e.g. Polish, Czech, etc) for the unaspirated version of the Pinyin "c" sound. German and Pinyin use the letter "z" for the same sound, etc.
Still, I agree with your main point. The Latin alphabet is not used like the IPA by most languages, even English has not preserved all the original Latin sounds as used by the Romans. And there's even precedent for some of the sound choices made in Pinyin that would seem completely alien to English speakers. For example "c" is used in all the Slavic Latin alphabets (e.g. Polish, Czech, etc) for the unaspirated version of the Pinyin "c" sound. German and Pinyin use the letter "z" for the same sound, etc.