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

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.

I think it may be because the unreduced pronunciations of a (/æ/ and /eɪ/) are front vowels. For comparison, "a" sounds in unstressed open syllables have been reduced to the "happy" vowel /i/ for some speakers in certain words; e.g. the -day week words or -aism words like Judaism and archaism. (See the question Which English words feature reduction of diphthongs like /eɪ/ to /i/?)

Alternatively, the identity of the following consonant might be relevant. The sounds /dʒ/, /t/ and /s/ are all coronal, and I think there is a tendency for vowels to be more front when they are adjacent to coronal consonants than e.g. when they are adjacent to labial consonants. I don't know enough about phonetics and phonology to be sure about this, though.