I agree with TN regarding the volume issues and the fact that the radiologists were there first. Although, TECHNICALLY speaking, cerebral angiography was pioneered by a Portuguese neurologist named Egas Moniz in the 1920s... though he won the nobel prize for his work in leucotomy, so some may consider him a neurosurgeon.
I'd say at this point in time, the radiologists are the ones who have "lost" the turf battle. The majority of neurointerventionalists are coming from neurosurgery and neurologists are growing in number. I feel as if the radiologists just didn't push for patient referrals, and surgeons and neurologists control the flow of most of these patients.
Now moving forward, I personally feel (probably have some bias) that the neurologists are the best poised to take over the field, as we are greater in number, and control the stroke patients. Stroke patients going to thrombectomy are arguably rising in number, where as aneurysms (classically the domain of neurosurgery) are tapering off, if not plateaued. There are further trials being conducted w/ intracranial angioplasty for ICAD, but we will see what happens with that in the future.
It's important to keep in mind that the surgeons have political power and can usually get whatever they want from administrators because they bring in $$$ for the hospital system. Neurology departments tend to lose money. As we learn more about other chronic brain disorders, and as more and more treatments become available, this may change, in the future, and we may command a more important role.
I strongly believe that if neurologists continue to be aggressive in this changing atmosphere of health care, we can strengthen our cause, and position ourselves well to follow in the footsteps of cardiology, but likely not in the same degree for the reasons that TN mentioned.