When holding a weapon or tool of any kind, skeletons will never pick up another bow, even with the CanPickUpItems tag. Dispensers don't work either. Skeletons with custom bows can still be obtained through commands, or by summoning a skeleton with no held item, as they will pick up a bow in this case.

Through testing, it seems this bug was introduced in 1.20.5, when item components replaced nbt