A folded dipole on its own is 300 Ohms impedance compared to a "straight" dipole which is 75 Ohms.
But when either of those dipoles are used as driven elements within a Yagi, the effects of the parasitic elements, primarily, the reflector and the first couple of directors will have a huge effect on the feedpoint impedance. It will lower that feedpoint impedance substantially. The 300 Ohms of a folded dipole can drop as low as 200 Ohms and a "straight" dipole of 75 Ohms can drop to as much as 20 Ohms.
In the case of the now 200 Ohms feed impedance of the folded dipole, this is quite an advantage, as a simple 4:1 BALUN transformer will match standard 50 Ohm coax to the feedpoint.
In reality tho, contrary to the comment in your first statement, straight dipoles are virtually as common as folded dipoles in Yagi antenna systems. Delta and Gamma matches are a common way to effectively impedance match 50 Ohm coax to a straight dipole within the Yagi.
cheers
Dave