I would think of it another way. Lets say your mom has a stroke, and shows up in the ED of your "stroke center" 2 hours after onset. The ED physician does her exam, then calls the on-call neurologist, at home. She says the stroke scale is "6-ish", but she isn't sure if the patient could understand her questions. The CT scan gets done, and per the musculoskeletal radiologist who is taking call that night, it looks negative. So the ED doc gives tPA. Patient doesn't get better. Goes to CTA which shows a right M1 cutoff. Patient then goes for STAT MRI with perfusion imaging. The ED doctor calls the neurologist back, wanting to know if she should activate the cath lab for MERCI clot extraction. It's now been 5 hours since your mother's stroke. The musculoskeletal radiologist reading the MRI thinks there is a big mismatch, but the neurologist can't tell if he is talking about the MTT or CBF imaging. Oh wait, now the ED doc thinks the patient might be improving. How the heck is the stroke neurologist supposed to appropriately decide these things over the phone?
I don't know about you, but if my mom showed up with a stroke to a "stroke center", then there had better be a stroke neurologist performing an exam and reviewing her imaging within an hour of arrival. There is just too much lost in translation over the phone, and too many decision points to make a phone conversation workable.
Stroke neurology has fortunately become a specialty where acute management can make a difference, and that management can take several forms. You can't run a trauma center without a trauma surgeon on-call, and in my opinion you can't run a stroke center without a stroke neurologist on call.
Where I work, there are 4 neurology residents in house every night, along with an in-house neurocritical care fellow, and the on-call stroke attending still comes in for every acute stroke within the time window for any therapy. It's the only way to really know what is going on.