noncontrast CT will be quicker and cheaper though.
This is an evolving and controversial area right now. I will briefly present the ultra-academic side of the argument that may become clinical reality in say 10 years. To state it simply, MRI has the possibility of taking over CT in the setting of acute stroke diagnosis. See for example:
http://stroke.ahajournals.org/cgi/content/full/36/9/1939
MRI Screening Before Standard Tissue Plasminogen Activator (rtPA) Therapy Is Feasible and Safe
The main argument as you stated is yes, CT is faster and cheaper in the setting of stroke. But, a MR stroke protocol can be done in 20 minutes at experienced academic centers. This is mostly because you want to get multiple contrast mechanisms while CT only gives structural information. Is the time and extra expense worth it? Maybe. The extra information allows you to make a better informed decision about how to treat the acute stroke patient. We know that MRI is more sensitive and specific to the cause of neurologic symptoms. That alone may make MRI worth it by more accurately selecting ischemic vs hemorrhagic strokes.
But, there's an additional, an upcoming reason why MRI may become standard of care in this scenario. If MRI allows us to better select patients that may benefit from rtPA therapy during stroke, it is likely to become the standard of care in hyperacute stroke scenarios.
See for example:
http://stroke.ahajournals.org/cgi/content/full/36/1/66
In this case they use the perfusion-diffusion mismatch to identify those patients that will benefit from rtPA therapy after the "3 hour window" that currently limits rtPA use. As you know, rtPA is a dangerous drug that can kill on its own due to the formation of hemorrhage. But, it is known that strokes still evolve after 3 hours in some patients and the 3 hour window probably is too restrictive.
See:
http://www.thelancet.com/journals/lancet/article/PIIS0140673604156924/abstract
Further, there are also many patients in which the onset of stroke is unknown (due to confusion or waking up with symptoms) and so it is not know how much benefit rtPA will give if given. Due to the danger of rtPA, it is often not given.
So if we had a technique that could select those patients whose strokes are still evolving after the 3 hour window or even better, could predict who still would benefit from rtPA even when time of onset is not known, the additional QALYs gained in stroke patients by selecting based on that information would likely make that technique cost-effective in this scenario. The PWI and DWI MRI may very well fill that niche (but it's not perfect either). A cost-benefit analysis of a larger trial is needed.
This imaging of ischemic penumbra, by any imaging technique, is an active area of research. It's what I'm finishing my thesis on now. We have a very large grant to study oxygen consumption techniques specifically for this purpose--to identify those who will benefit from rtPA therapy.