Chief John Ross moved to Philadelphia because he had family living there and it was close to Washington. He attempted in vain to gain assistance and more military support for the "loyal" Cherokee.
Not only was Ross too old and frail for command in the field, the US Army required white officers over the Indian regiments. The highest ranking Indian in the Union Army in the Indian Territory was Lt Col Lewis Downing of the 3rd Indian Home Guard (aka "the Pin Regiment").
The Cherokee Treaty Party was a minority party who signed the illegal Treaty of New Echota in 1835. The Ross Party, who was by far the majority, blamed them for the Trail of Tears. While I believe the Treaty Party took the right course for the long-term good of the tribe, they knew when they signed the treaty they would be assassinated.