Hmm...I've read all 12 of the American adult Indiana Jones novels and of course, seen all 3 of the Indiana Jones movies numerous times (as well as The Young Indiana Jones Chronicles, for that matter) - Can't wait for Indiana Jones IV!
I guess I can see how Indiana Jones was sent on secret missions some in the books (particularly Indiana Jones and the Sky Pirates...also Indiana Jones and the Philosopher's Stone, as you said...)
But Indy was sent on missions in the movies too- wasn't he sent to find out what the Nazis were up to at Tanis in "Raiders of the Lost Ark"? Top government men came to recruit him then and he went on to snatch the Ark of the Covenant away from the Nazis. He was again thrust into adventure in "Indiana Jones and the Temple of Doom" when the townspeople recruited him to find the Sankara stone and rescue their children.
And Indy was recruited by Walter Donovan to find the Holy Grail in "Indiana Jones and the Last Crusade" (though he mainly went only to find his father)
I can see how Indy might be recruited by the government for his resourcefulness and knowledge and skill as an archaeologist. I wouldn't want him to necessarily be Nick Fury, Agent of S.H.I.E.L.D.- but it is kinda cool to see him go on missions for the FBI, CIA or Her Majesty- especially considering James Bond is his daddy!