I notice that most small power transformers (single-phase, 100-240V, 50hz/60hz, laminated core) are shell-type, with an EI-shaped core. But as I understand they have relatively low efficiency due to much of their surface area being exposed; not wound with wire. There is already much information on the web about the advantages and disadvantages that toroidal cores have over EI; being entirely covered by windings improves efficiency, but they are intolerant to DC offset, harder to wind because they must be wound when the core is fully constructed (as opposed to wrapped around a bobbin before the core is built around it), and mounting is less straight-forward.
But I want to know why EI transformers are often chosen over core-type core constructions such as U-core, C-core, or L-core? These are presumably more efficient, due to having more surface area covered in winding (even if less than toroidal). They seem to have the same construction advantages of EI transformers, such as being wound on bobbins before the core is assembled, mounting methods which are almost as convenient as EI, and laminations for some designs can be cut from a rectangular sheet without significant waste (like EI-cores and toroidal cores). Also, the air gaps reduce the effects of saturation.
So what are the disadvantages of these core-type transformers that leads to them being less common than EI-cores?