I think the premise of the question is problematic. Pinyin was not created to teach the sounds of Mandarin to second language learners, so the question of why it is in a certain way that happens to be problematic for second language learners does not have an answer.
For native speaker, there's no problem using "i" for three completely different sounds, because the syllables involved are in complementary distribution, meaning that they never overlap and you can always know which one it is from context. This is similar to how ü drops the diacritic after j/q/x, which is also a problem for second language learners, but not for native speakers.
Why? Because they learn to hear and speak the sounds before they learn to write them. The problem you describe is partly caused by the fact that students learn pronunciation from writing.
The angle of approach shouldn't be "how is this letter pronounced", but rather "how is this sound written down". The problem you bring up can be avoided by focusing on the pronunciation first and then teaching how that pronunciation is written down.
Finally, if you teach the finals as units, things will become even easier. There's no good reason to focus on letters in Pinyin. What about a that is used to write maybe four different sounds? Or e?
There aren't that many finals, so teaching them as whole units is possible.