1. The Eagles are not much interested in the affairs of those on the ground.
They go home because they have no need of ground level allies or associations.
However, the politics of accepting gifts is very important in Tolkien - they
should not really be turned down.
This passage, however, tells the reader a more important thing - Dain is not
trying to hold on to the gold. He is giving it freely and generously. The
Eagles have no possible claim on the gold (much less than the people of Erebor
or Dale). Dain has no taint from the dragon, or from gold in general. Big
textual signpost saying - Dain will be a good king!
2. Wistful nostalgia is Tolkien's natural mode. Wishing for impossible things
drives so much of the narrative of the Hobbit and LOTR (and of course Niggle
and Smith). Much of the time it is where the good in people comes from -
Faramir wishing for Gondor to be a Queen among Queens, Sam wanting to see the
Elves, Bilbo's need to see the world again more than his need to hold onto the
ring. It's a very important theme, I
think.