If Mary had her mind on a family social connection that could fast-tract Edmund in the military or politics it seems she would have said something of the kind. Mary does not have very much family to rely on. There is no evidence Admiral Crawford has useful connections or patronage to bestow in any profession other than the navy. Considering how the Admiral has treated Mary why would she have any confidence he would do anything to promote her happiness? Dr. Grant seems equally limited unless there is some text to suggest otherwise. It is not impossible they have useful connections beyond their professions but I do not remember any text that suggests it. Henry seems too young and unfixed to have formed useful connections on his own. Of course I could be entirely wrong. (:D)