That would be the opposite of consistency; i is the front vowel and ı is the back one.
Note that the vowel /i/ cannot umlaut, because it's already a front vowel. The ï you cite comes from French, where the two dots represent diaeresis rather than umlaut. When umlaut is a feature of your language, combining the notation like that isn't likely to be a good idea.
Turkish i/İ sounds pretty similar to most of the European languages. Italian, French and German pronounce it pretty similar. Also removing umlauts from the other two vowels ö and ü to write o and u has the same effect as removing the dot from i. It is just consistent.
No, what I mean is, o and u get an umlaut (two dots) to become ö and ü, but i doesn't get an umlaut, it's just a single dot from ı to i. Why not make it i and ï? That would be more consistent, in my opinion.
I guess the aim was to reuse as much of the standard Latin alphabet as possible.
A better solution would have been to leave i/I as they are (similar to j/J), and introduce a new lowercase/uppercase letter pair for "ı", such as Iota (ɩ/Ɩ).
This was shortly after the Turkish War of Independence. Illiteracy was quite high (estimated at over 85%) and the country was still being rebuilt. My guess is they did their best to represent all the sounds while creating a one to one mapping between sounds and letters but also not deviating too much from familiar forms. There were probably conflicting goals so inconsistencies were bound to happen.