Why do these units have a shoot value? Clearly they have no missile weapons and I don't see Throw Stones for them, even in the errata.

In the early editions of SBG, models can pick up weapons and use it after slaying an enemy. I guess they just left it on the profile.

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Can't see an eagle using any weapon. I thought maybe it was for dropping boulders or picking up enemies and dropping them like in the movie. My boy isn't impressed that the eagles don't have a special rule for picking up wargs

They sure do. Eagles are monsters, which means they can Hurl (see the Hobbit rulebook) wargs. The rules are found under Brutal Power Attacks rather than on the profile.

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