Now, let me share with you all my point of view about how Moegi awakened the wood release. We all remember the Senju clan, from which the first and the second Hokage were from. And more importantly the first wood release user Hashirama Senju was from this clan.
But due to some mysterious reasons, the club vanished after the death of the second Hokage, Tobirama Senju. The most famous theory and the most believable one is that in order to increase the manpower of Konohagakure, Senju clan intermarried with numerous clans. Having the bigger picture of the betterment of Konohagakure they decided not to care about their surnames that much and let their children use their mother’s surname. Tsunade Senju is the only exception.