I don't know that it is wise to develop multiple groups of armed security teams ... each may have different objectives, different training, different procedures, different schemes of command & control, different missions. I think that whichever organization is responsible & accountable for security of the airport facility should organize, train, equip and control the armed security personnel. That squares with the time-tested military principle "unity of command". I think this is not TSA, but the airport owner/operator. Of course all involved should work together to coordinate their activities so that we have a comprehensive & integrated system.
One thing I did find interesting on the TSA website ... TSA identifies many layers of security, but I don't see local airport security or law enforcement anywhere on the graphic. That is a gap that should be closed.