i guess it is because in this sentence we are talking about the president of republic.
republic = cumhuriyet and president = başkan --> cumhuriyet başkanı --> cumhurbaşkanı
The word for president is "cumhurbaşkanı". That's the word itself. It's not like it was "cumhurbaşkan" and then it received a suffix. It's always with an ı.
It isn't really correct as this is a catch all term (for example, this is both the title of the President of Turkey and the President of the United States).
It's late but, while it isn't necessarily wrong, I think most people would use "Başkan" as in "Amerikan Başkanı" or "Amerika Birleşik Devletleri Başkanı" for the President of the US.