See my response (number 4) to the MO question:

http://mathoverflow.net/questions/23222/exponential-map-and-covariant-derivative

There is a Math Review article by Kuranishi there of the paper "Sprays" by Ambrose, Singer, and Palais (in which sprays were first defined). I think the approach taken there (and described in Kuranishi's review) is pretty much what Dan Ramras is asking for.