They would only have to trace the pathways if Bitcoin and shut down the relative few that use them.
That's... not exactly how blockchain works.
In a basic blockchain, every participant in the chain has a register of all transactions within the chain. When something changes on one node, every other node is updated immediately with that change. It's an indelible register. The whole point to blockchain is to make fraud impossible. If one node or even a lobe of the chain goes down or becomes unavailable, the rest of the chain continues to function and updates the missing nodes when they return. It's a multi-master system.
Truly the only way to ban Bitcoin would be to completely collapse the Internet into so many shards as to make the blockchain impossible to reconstruct.