You may not have mentioned this because it seems so obvious, but I think it's useful to note this reduction only occurs in unstressed syllables—primary stress (as in the U.S. pronunciation of ballet) or secondary stress (as in the first syllable of aviation) seems to prevent it.
The English diphthongs that end in a front offglide are /aɪ/, /ɔɪ/ and /eɪ/. Some analyses do not recognize /eɪ/ as a true diphthong, and treat it as somthing like /e/ instead.
For /aɪ/, it’s hard for me to evaluate if reduction to /i/ has ever occured. In spelling, both “i” and “y” can represent either the diphthong /aɪ/ or the reduced happY vowel /i~ɪ/.
So if there were any cases of historical reduction of /aɪ/ to /i~ɪ/, it would be hard to deduce this from the spelling of words.
As far as I can tell, there isn’t much evidence of a /aɪ/ to /i/ shift happening historically, but there is some evidence in rhymes that suggests that before the Great Vowel Shift was complete, some word-final vowels went from [əɪ] (the vowel that corresponded to past /iː/, and in mosts contexts future /aɪ/) to [i~ɪ]. See these questions for more details: Was the pronunciation of “symmetry” different in the past?, Rhyming conventions of Early Modern English. I suppose an alternate explanation could be that shortening occured before diphthongization, and the apparent pronunciations of words like symmetry with diphthongs were eye-rhymes or poetic license.
The diphthong /ɔɪ/ is pretty rare, relatively speaking, so I don’t know of any examples of possible reduction of this vowel to happY.
Reduction of unstressed /eɪ/
The “long a” sound /eɪ/ has, as you noted, apparently often been subject to reduction to happY in fully unstressed syllables. (It seems to be more common in post-tonic syllables than pre-tonic ones; I haven't found evidence of it affecting the unstressed "a"s in Laocoon, aorta or aedes/Aedes.)
But there aren’t many words where this is relevant; aside from the ones you list, I was only able to find the following. Many of them have alternative pronunciations with unreduced /eɪ/, or other alternatives.
Common nouns. So far, I've only found:
-aïsm words: archaism /ki.ɪz.m/, /keɪ.ɪz.m/, Judaism /di.ɪz.m/, /deɪ.ɪz.m/, /də.ɪz.m/, /ˈdeɪ.ɪz.m/, Hebraism /breɪ.ɪz.m/, /bri.ɪz.m/(apparently less common, but in Random House dict.).
Apparently NOT Pharisaism /seɪ.ɪz.m/ (dictionaries assign the pronunciation with /si.ɪz.m/ to the alternate spelling "Phariseeism", although I don't know how they could tell which word someone is using in a spoken context)
some -et words from French, in British English: crochet (/ˈkrəʊʃeɪ/, /ˈkrəʊʃi/) and croquet (/ˈkrəʊkeɪ/, /ˈkrəʊki/). I don't think the reduced pronunciations are very common. In American English, these are generally stressed on the second syllable, so reduction is impossible. But ballet is not an example according to the OED.
Biblical names: Ishmael /mi.əl/, /meɪ.əl/, Ephraim /reɪ.əm/, /ri.əm/, /rəm/,
Sinai /naɪ/, /neɪ.aɪ/, old /ni/ (see Wells),
Not examples: Sarai /raɪ/, /reɪ.aɪ/ (so far, only found this last one in Dict.com/Random House Dict.)
Classical Greek names: Nausicaa
Not examples: Pasiphae and Danae look similar, but they seem to have reduced pronunciations where the vowel is commA/schwa, not happY, possibly due to dissimilation with the following vowel.
The following Biblical geographical name apparently used to have “a” pronounced this way, but now no longer does: Canaan /nən/; formerly /niən/ or /njən/ (see Wells)
And the following names/proper nouns might possibly have had “a” pronounced this way a long time ago, but I haven’t found any evidence of it: Balaam /ləm/, Pharaoh, Michael, Isaac
There is another interesting thing about some words spelled with “a” before another vowel. Some speakers use /aɪ/ in some of these words instead of /eɪ/ or /i/, although this pronunciation is somewhat stigmatized. This may be influenced by analogy with the word naive. The words I know of where this may occur are: Israel, Naomi (see Wells).
With words ending in the letter "e", such as karaoke and linguine, it's a bit difficult to tell whether the pronunciation of this "e" as /i/ is due to reduction of /eɪ/, analogy with other words spelled with "e" like epitome or hyperbole, or simply difficulty that English speakers have with hearing unstressed word-final front vowels like /e/ as sounds that are distinct from the "happY" vowel /i/. (Other possible factors for linguine are analogy with other Italian pastas that have names ending in /i/, like spaghetti and vermicelli, and dialectal variation in the pronunciation of word-final vowels in Italian, as described in this ELL answer by CoolHandLouis, that might have influenced English pronunciation.)
The word always may have a reduced final vowel; since it is in a closed syllable, [ɪ] is possible instead of [i].
"Canaan and Sinai", John Wells’s phonetic blog