That may be true, and it's a good point to remember. But you are still making what is essentially an argument ad hominem against the Cherokee declaration, rather than accept its face argument, that the Cherokee had suffered mightily at the hands of the U.S. Government. They had also had bloody differences with white settlers even in colonial times, per the comment above about the Cherokee's and other tribes' cooperation with the British, which kept the white settlers in the three prerevolutionary counties of Tennessee forted-up and unable to expand for ten long years, and their population stagnant thanks to disease and malnutrition. That will have been a bitter memory on the other side.
Bottom line, the Cherokee had had concrete experience with the operational policies of the Government and the kind of people who drove them that illuminated their own best interest and guided their choice of allegiance in 1861.
Or is that not a fair statement?
Frankly, I don't think the tribes would be united, or even thoughtful, enough to have a reasoned political position such as you describe. I think that they took part against the federals through warrior passion, or from a sense of feeling invaded, than through a rational philosophy.
They probably took arms because their neighbor took arms, and nothing more complicated or theoretical than that.
There is an interesting essay today on "Cold Mountain" on National Review Online--this is the Civil War that I heard about from my own ancestry. Quite a resonation, this particular essay.