As someone who lives in a state where county-wide or "central" dispatch is the rule rather than the exception, to me the biggest issue is a single PSAP for a particular county is politics. You're likely going to have at least one L/E or fire/EMS agency that's going to insist on either maintaining its own dispatch center or even it's own radio system, compared to the rest of that county. Breaking down those barriers and "kingdoms" is going to be the critical first step in forming a county-wide dispatch.
In the Philadelphia area, for the most part, most of the counties have a central dispatch point for all PD/FD/EMS agencies (Bucks, Lancaster, Lebanon), though counties still have a few self-dispatched muni PD's (Chester, Delaware, Montgomery, Lehigh, Northampton, and Berks). For the most part, these dispatch centers are county-run centers that are under the county's Department of Emergency Services/Office of Emergency Management (whichever is the term) as opposed to a third-party (which seems to be how you're describing "Central Dispatch"; if I mis-read that, then I apologize).