The /ˈɛryən/ pronunciation is just a result of English phonology processing a foreign borrowing that starts with the letters AR.
Aryan is a borrowed word in all languages outside the Indo-Iranian subfamily of Indo-European. The rest of the world pronounces it as some variant of [arjan], which comes, as noted, from Sanskrit ārya /a:ryə/ 'compatriot'. Therefore, /'aryən/ is a perfectly acceptable English pronunciation, and the only acceptable one when using the term in its modern Indian sense.
Any use of Aryan (outside scare quotes) that refers to Germany or white racism is a result of romantic interpretations of 19th century German linguistic scholarship (e.g, Grimm's Law), which unearthed the prehistory of the "Indo-Germanic" (as I-E was then called, from names of its Eastern- and Westernmost families) languages. It was all very exciting, apparently. See also Wagner, Mad King Ludwig, Neuschwanstein, German Empire.
The AHD of IER says that Skt ārya comes from the PIE root *aryo- 'Self-designation of the Indo-Iranians'; other descendants of the same root are Iran and, surprisingly, Eire -- Celtic languages sometimes retain PIE roots that are otherwise lost in the Centum group.
None of these are English words, and so English treats them the same way it treats all borrowed words -- it changes the pronunciation until it tastes right. That's all.
Edit:
I almost forgot, another reason to pronounce Aryan /'aryən/ is because Arian /'ɛriən/ usually refers to Arianism, a very important variety of Christianity that was the religion of the Ostrogoths in Italy, and the Visigoths in Spain. The only Gothic texts known to exist are translations of various parts of the (Arian Christian) New Testament.
The pronunciation of "get" is indeed a very regional thing. I believe the "prestige" pronunciation in the USA is "/gɛt/". However Southern American English shifts a lot of vowels up, including shifting "get" to /gɪt/. The related African-American Vernacular English has the same feature. That means the predominant accents in both the Southeast quarter of USA and in most of its larger urban areas uses /gɪt/.
It's even a bit more complicated than that though. My own accent (I believe a mixture of Midlands and Southern Midlands) pronounces the word slightly differently depending on where it appears in the phrase. If it's in the middle, it is more likely to be closer to "/gɛt/", while at the end, it's more like "/gɪt/".
For instance, when my kids were in daycare, a popular mantra from the daycare workers was:
"You get what you get, and you don't throw a fit."
The way this is pronounced, the two phrases rhyme. However, the two "get"s don't quite rhyme with each other.
Best Answer
The traditional English pronunciation of word-final "ae" is /iː/, as in "algae". You can see examples of this in F'x's answer to Pronunciation of words ending with “‑ae”. There is nothing special about the word "formulae" that I know of that would cause it to deviate from this pattern.
The pronunciation /aɪ/ is derived from the "restored" English pronunciation of Latin. Basically, sometime around the end of the 19th century (I don't have a great handle on the exact timeline, sorry), people stopped using the traditional English pronunciation of Latin to teach Latin as a language.
In its place, a new pronunciation system was developed for English students of Latin to use that was meant to approximate the reconstructed sound system of Classical Latin. As with the approximate "pronunciation guides" for living languages that are often used in low-level language teaching (e.g. pocket language guides), there are some notable differences between the typical "restored" English pronunciation and how we think the Romans actually pronounced things. "Restored" English pronunciation could be thought of as reconstructed Latin spoken with a more-or-less strong English accent. (E.g. English speakers using "restored" pronunciation often diphthongize the vowels "e" and "o" in certain contexts to something like [eɪ] and [oʊ], which is not a feature of reconstructed Latin pronunciation.) It's not clear to me how intentional the gap between "restored" English pronunciation and actual reconstructed Latin pronunciation was supposed to be; it seems possible that some idealistic teachers supposed that they could actually teach their students to speak Latin like the Romans, but the typical weaknesses of foreign language instruction led to the widespread use of more English-phonology-based pronunciations by students.
In the restored pronunciation of Latin, Latin "ae" is equated to the English vowel phoneme /aɪ/, as in the word "price". This is because there is a fair amount of evidence that Latin words spelled with "ae" in the Classical period were at one point pronounced with a diphthongal sound something like [aɪ̯] or [ae̯] (the use of the spelling "ae" in Classical Latin suggests that the pronunciation of the final element was somewhat lower than [i̯]). We know this partly based on etymology, and I believe there are also descriptions of the phonetics of Latin sounds in some Classical texts. The English "price" vowel in most accents is a diphthong in this general area (the phonetic details of the "price" vowel of course vary between accents, and depending on the phonetic context).
Even in the Classical period, there is evidence that some Latin speakers pronounced "ae" as a monophthongal vowel with a quality similar or identical to Latin "long e".
By the time of Medieval Latin, I have the impression diphthongal pronunciations of "ae" were entirely extinct. According to Wikipedia, in Medieval Latin writing, "ae", "e" and "ę" were all used to represent both Classical Latin "ae" and Classical Latin "e". In the traditional pronunciations of Latin used in Continental Europe (e.g. "Ecclesiastical" or Italian traditional pronunciation of Latin), "ae" is pronounced as a monophthong with a quality somewhere in the area of [e~ɛ] (I don't remember the details of vowel quality for different systems, and I think there is some variability or uncertainty).
This is the source of the traditional English pronunciation of "ae" as [iː]: due to the "Great Vowel Shift", [eː] and [ɛː] turned into [iː] in English pronunciation.
In my experience, most prescriptive sources that bother to take a stand on [iː] vs. [aɪ] for "ae" in modern English words are more in favor of [iː], although my experience may not be representative. One relatively prominent strand in the prescriptive tradition is an emphasis on consistency, and pronouncing "ae" as [aɪ] in words like "formulae" is inconsistent with the established pronunciation of this digraph in a number of words (like "Caesar" and the aforementioned "algae"; also, in British English, words like "anaemia" and "anaemic"). Furthermore, the use of a "restored" pronunciation for "ae" introduces inconsistency not only between words, but also within words: "formulae" is pretty much always pronounced with a palatal glide /j/ after the /m/, even though the restored pronunciation would call for pronouncing the "mu" in this word as /mʊ/ with no glide after the /m/.
A third minor pronunciation for "ae" in English exists, [eɪ]. This represents an English approximation of the [eː]/[ɛː] value that I mentioned occurs in Continental European traditions for the pronunciation of Latin, like "Ecclesiastical" Latin. There may also be some minor influence from spelling, since "long a" in English is pronounced as [eɪ], and word-final "e" often indicates a "long" pronunciation of the preceding vowel. I don't think [eɪ] for "ae" is endorsed by many prescriptive sources (in fact, I don't know of any).
As far as I know, none of the various pronunciations of word-final "ae" in English words is "universally" embraced by educated individuals. Most people don't have the occasion to pronounce words like "formulae" that often.