Mu-metal can have relative permeabilities of 80,000-100,000 compared to several thousand for ordinary steel. In addition it has low coercivity and magnetostriction resulting in low hysteresis loss. Its magnetic properties are no better than other high permeability alloys such as Permalloy, but it is more ductile and workable.
Magnetic shielding and magnetic cores rely on the same magnetic properties so you shouldn't be getting overly confusing thoughts as to why Mu-metal might be good for shielding but no good for cores (as far as magnetic properties are concerned). It's probably just not used for cores since ductility is not as important with cores as it is with shielding. Permalloy (and other NiFe purpose-designed alloys) are just as permeable and cheaper since that last 5% is nickel instead of molybdenum which is about 1/3 the cost.This question may be a little bit difficult to answer, but would mu-metal, make a good electromagnet core? I know it has a super magnetic permeability, but it's also used for magnetic shielding.