pronunciation - Why does the letter "a" correspond to /ɪ/ in words like "image", "private" and "surface" (American English)?

In American English, in words ending with -age, -ate and -ace, the ‹a› correspond to /ɪ/ (short i). Examples:

image, village, damage
private, senate, separate
surface, preface, palace

(It should be noted that dictionaries do not always agree about the pronunciation, and some use /ə/ instead of /ɪ/ for some of the words).

In all of these words the last syllable is unstressed. In unstressed syllables vowels may be reduced to schwa or to a short vowel of a similar quality, like /i:/ to /ɪ/ and /u:/ to /ʊ/. But at the words above, the underlying vowel in the unstressed syllable is either /eɪ/ or /æ/, so it is unclear to me why the reduction is to /ɪ/.

Answer

Why do you assume that the “underlying vowels” of these suffixes are /eɪ/ or /æ/? All of your example words are Old French or Latin in origin. In Middle English they would have been pronounced with [aː] or [a], which was later reduced to [ə].

The change from [ə] to [ɪ] has little to do with reduction, per se, but rather allophony—the sounds simply exchange in unstressed syllables for many speakers. From “Stress and vowel reduction in English”:

In some dialects of English there is a distinction between two vowel heights of reduced vowels: in addition to schwa, there is a distinct near-close central unrounded vowel [ɪ̈] (or equivalently [ɨ̞]). In the British phonetic tradition, the latter vowel is represented with the symbol /ɪ/, and in the American tradition /ɨ/. An example of a minimal pair contrasting these two reduced vowels is Rosa’s vs. roses: the a in Rosa’s is a schwa, while the e in roses (for speakers who make the distinction) is the near-close vowel.

Among speakers who make this distinction, the distributions of schwa and [ɪ̈] are quite variable, and in many cases the two are in free variation: the i in decimal, for example, may be pronounced with either sound.

This is true for all of your example words as well. Dialects where the sounds do occur in free variation are said to have the weak-vowel merger.