Isn't it the case that a node seeing a second spend attempt will not broadcast that transaction? In which case a double spend will look like a single spend to most of the network, since they will only see one of the two transactions.
This is correct, a node won't forward a transaction that it considers invalid, and by seeing another transaction that spends those same inputs, the second transaction is invalid. It probably wouldn't need 15 seconds even if the network were huge. Transactions propagate very fast.