This is an excellent question. I'd argue that we don't need to reinvent the wheel. I'd also argue that this will not be an overnight fix. It will take time and will require patience.
As loathe as I am to make this comparison look at nursing. Nursing used to be a diploma or certification only educational program. Now it's at least an associates program. It some places nursing jobs are only available to BSN applicants. It didn't happen overnight for them. It won't happen overnight for us.
For EMS I think NREMT has sort of started this process. Paramedic programs need to be accredited (as of 2013) by the Commission on Accreditation of Allied Health Programs. This is a good first step. It's been this way for six years now. The next step might be to require accreditation at a degree awarding institution by a certain date. Then require new those earning new certifications after a certain date hold at least an associates degree. It'll be a multi-step process undertaken over years to make the change. There will probably be some grandfathering in of older providers and/or a grace period during which providers will need to complete a bridge course of sorts (similar to the RN to BSN programs that are out there). There will be push back from old school EMS-ers (No degree is going to help me start that IV any better!) and fire departments (What do you mean our medic mill that pushes out paramedics from a condensed program only so they can ride an engine isn't good enough?). Like old school nurses and old school nursing diploma programs, they will lose.
As to why there is a shortage in some areas you have to look at a larger picture. Is there a shortage of just EMS-ers? Or is there a shortage of everything else? RNs? Docs? PAs/NPs? Access to basic services? Why is that? In a lot of cases because it's rural and there's little incentive to undertake the effort. Stop relying on the volunteer aspect which, ultimately, cheapens us all and accurately value the services provided by educated EMS providers. Do this and I think you'll see a change in the shortage. (Maybe not fix entirely, but certainly lessen the shortage.)
Under no circumstances am I arguing this will be easy. It won't be. There will be a lot of push back from a lot of entrenched special interests. Until we fix education, however, nothing will change. Fix education, align ourselves as legitimately educated, degreed, licensened *MEDICAL* providers and not some haphazard add on to another public safety agency, and every problem currently facing EMS will go away.