Blame the press:
Washington Post reporter asked Trump if he was prepared to put the birther issue to bed (since Trump was said to have led the Birther issue in 2008).
Trump responded not at this time ... because he had already planned to do so in his speech the next day.
Washington Post then ran a headline that Trump refuses to admit that Obama was born in the US
Clinton campaign then jumped on it.
I think there was a first step you didn't mention.
Clinton campaign contacts Washington Post - requests the birther issue to be brought up with Trump.