Why do Flyback Transformers have C-I cores?
Assuming you're talking about an old-school CRT flyback transformer and not a common switchmode flyback transformer (which are generally not C-I), I think that it's a cost optimization based on all the insulation required for reliable operation at high voltage- since the secondary coil is bulky relative to the amount of copper, a lot of ferrite material is required to make a closed magnetic circuit that clears the coil, so it's cheaper (less total volume of ferrite) to have only one leg closing the magnetic circuit.