First, you're wrong. First ladies have the power of access to the president, and the bully pulpit. Their power is entirely based on social respect.
So the things first ladies push are only tangentally connected to the president's agenda. If they push his agenda too closely, they can and will be attacked by the policy's opponents, and that tradeoff is never worth it.
So first ladies push less controversial things. Nancy Reagan had a generic anti drug abuse message. Laura Bush, Literacy. Michelle obama, better school lunches. Betty Ford, rehabilitating addicts.
Hillary by comparison pushed Hillarycare. A gigantic nationwide health care program, and got blasted for it, as she wasn't an elected representative. Supposedly it made her so assblasted that she got herself implanted into the senate and carved her own road to the white house, but that's neither heir nor there.
However, the MOST important role that the first lady has is in soft diplomacy. Running social, diplomatic and hospitality events for visiting diplomats, heads of state, their families, ect.
Pic related to /co/ and first ladies: Named after Dolley Madison, president Madison's wife.