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.
Best Answer
Great question! Save for the oral echo, there's no necessary relationship between gerontology and ontology. The t comes from the genitive case.
Here is the Ancient Greek Wiktionary entry for γέρων (geron). Note that, unlike English, Greek has several noun cases, including the genitive γέροντος (gerontos), the dative γέροντῐ (geronti), and the accusative γέροντα (geronta).
Now, think about the meaning of gerontology you suggested. It is the study of old men. In another language, old men might be in the genitive or dative, indicating some relation to the head noun. The root being adapted into English may come from this form. So in that case we would have geront- plus logos, which after adjusting each to fit English paradigms for Greek roots (logos to -logy with -o- as a combining vowel, gerontos or geronti to geront-) results in gerontology.
Odontology behaves in a similar way, with the root being ὀδούς (odous) but changing in the genitive form to ὀδόντος (odontos). (See also orthodontics and other toothy variants.)
More generally, these are examples of classical compounds.