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.

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.