No it wouldn't collapse, but by every precedent in the franchise all the military, political and economic hubs of the Federation center around Earth. Starfleet especially is VERY Earth dominated.
I've already argued that the rest of Starfleet and the Federation probably wouldn't go along with this in this thread, but it is reasonable to assume that by commanding Earth you have some authority to exert with the rest of Starfleet and the Federation. I already compared Leyton with the July 20 plotters in Germany who surmised that if they eliminated the Head of State and controlled Berlin they'd control Germany. That failed too.