Mohism wasn't actually the largest school, as none of the major states adopted their thoughts. However, they are the most powerful school if one don't count the support from the states.
Perhaps the unification process can happen longer than OTL. If a series of reforms sweep over the Warring States following Shang Yang's reforms, the period might be able to be prolonged. Then, Qin may possibly be able to have a war against Zhao, which would cause great casualties on both sides...
The possibility of they sending atheletics may result from more connections and respect from the West, which meant the possibility of large-scale mutually beneficial trade in the 18th century, and the use of foreign firearms earlier.