Learn English – We have “Here” and “There”, why not “Hen” and “Then”

It certainly wouldn't have been impossible for some alternate history version of English to have ended up with those abbreviations. However, we need to consider the things that lead to abbreviations happening at all.

The need for them has to be relatively common and they have to actually shorten significantly. If neither of those is true, nobody will bother to create the abbreviation.

To take root the need has to be relatively widespread. There is also a tipping point effect; up until a certain point the greater likelihood is that the abbreviation will just die out, but beyond that point so many people are using it that it becomes self-sustaining (the same as with any other term).

For the same reason, an abbreviation is less likely to gain currency if its need is already adequately filled by another. (A notable exception would be the many recent abbreviations referring to laughter, such as LOL, ROFL, PMSL etc. but there is a strong degree of deliberate play there, which encourages more permutations than would exist otherwise).

Now, both i.e. and e.g. are most often used in relatively formal writing that is putting forward an argument, or otherwise expositionary or scholarly.

At one point, such works would not be written in English, but in Latin. Only English people could read English for one thing, while any educated person in Europe could read Latin, especially considering the link between religion and higher education that once existed. Bede in the 7th and 8th Century wrote all his important works in Latin. Chaucer is "the Father of English literature" because he bothered to write in English at all, when most wrote serious works in Latin or French, and even he wrote his non-fiction in Latin. In the 15th Century Latin grew in secular use (ironically, the same Protestant scholars who rejected Latin in the prayer-book and the Bible, were particularly fond of it in the sciences, including most English scholars), and so scholarship continued to use it heavily. It began to decline around the start of the 18th Century (consider Newton, writing his earlier important works in Latin, his later in English), but continued to have considerable academic use until the end of the 19th.

And since all these people were writing in Latin, they would of course use i.e. and e.g. in the contexts they most come up rather than t.i. or f.e..

Now, any such academic writer would have a strong knowledge of the more commonly used abbreviations, along with scribal abbreviations, which are a form of abbreviation that combines letters and from which we get #, $, £, %, &, ‰, lb, &c. §. and indeed pretty much all of the oldest abbreviations used in English (etc. et al. ca. cf. ibid. op cit.) along with the practice of doubling for plurals (pp. for "pages", SS for "saints", §§ for "sections", etc.).

Note that while Latin was used throughout Europe, it had regional forms the same as English does now, and all the more so with abbreviations. For example, while both & and ⁊ were found throughout much of Europe, both being abbreviations of et, they survived in different languages (& used in quite a few as well as English while ⁊ is now pretty much only found in Irish and Scottish Gaelic in which & is not found).

These Latin-using scholars both used these abbreviations with which they were familiar both when they came to write in English, and if they came to teach English writing to others (and scholars was the pool of people from whom the best teachers were hired, after all).

For this reason, the abbreviations came to be known by literate English-speaking people even if they didn't speak Latin themselves. By this point, i.e. was almost as much a part of written English in a particular register than dog or cat was, and almost more a part of it in that register than that is!

It was at this point, when most people writing English in the register in which i.e. is used, that potentially t.i. could have taken over. But why would it? Why would people suddenly start using t.i. when the perfectly good i.e. that everyone knows would do a better job, because everyone knew it?

Latin also survived in different ways in other languages, such as the example of ⁊ in Irish and Scottish Gaelic I gave above. For this reason those expressions of Latin origin known to other speakers of European languages won't overlap fully with those used in English.

In the journey from Old English to what we write today, the ash (Æ) tended to metamorphose into a simple E and various "ae" forms got reduced to just "e": Ælfwyn became Elvin, Æthelræd became Ethelred, aether and aesthetic became ether and esthetic (except when @Cerb spells them), and so on. The distinction was simply planed off over the centuries. When there was no need for the superfluous Æ (because its sound was rendered with a single letter) it got dialed way down in frequency. The same thing happened, more or less, with the thorn (Þ, þ) and eth (Ð, ð) characters, because the th digraph supplanted them.

An even more obvious influence involved the printing press. In the early days of typography, fonts were imported from Germany and Italy, and those countries did not use the oddball English characters, so substitutes had to be found. "E" substituted quite nicely for the ash, and "Y" for the thorn (as we see on the signs in front of all those cutesy Ye Old Whatever shops).

Elaboration

Asked for citations, I lazily looked to the Web first, but real scholarship in this matter is difficult to Google. Here are some not-stringently-academic citations, together with a disclaimer.

The thorn was particularly popular as a sign for 'th' in Medieval English, but with the advent of printing came a problem. There was no thorn sign in the printing fonts, as they were usually cast outside of England. So, since the sign for thorn slightly resembled the lower-case 'y', that's what was substituted.

The thorn continued to be used, but printing caused its eventual demise from the English alphabet. As mentioned earlier, lingering proof of its existence hangs on in the outmoded 'Ye'.
Thorn — Missing Letter of the Alphabet (reposted with correct glyphs here).

Ultimately, the letter was abandoned when printing began to streamline the alphabet and eliminate unnecessary letters. Æ was separated into AE, and the language moved on. However, you can still find ash used stylistically in names like Encyclopædia Britannica and ÆON.
Mighty Markup.

Disclaimer: I feel it only fair to point out that the reference books I have at hand (printed versions, so no linky-link), suggest that the ash (or æsc in OE), was pretty much gone by 1250 due to the influence of Norman French. This was a couple hundred years before the invention of the printing press, so we cannot accept that as the proximal cause. Still, Gutenberg almost certainly put the nail in the coffin of that and the other oddball characters (including wynn and yogh — look those up for your amusement and edification sometime).