They have the testimony of David Pecker who outlined in detail the "catch and kill" conspiracy between Trump, Cohen and the Enquirer. They were to silence those with dirt on Trump and promote stories to have his opponents, whether true or not, until AFTER THE ELECTION.
They have the testimony of Hope Hicks that Trump told her that it was "better" that this stuff came out after the election than before it.
If Trump's PRIMARY concern was that his family not find out, why was it OK to publish the stories after the election?
Melania and his family weren't even mentioned until the day the story was published in the Wall Street Journal, and Trump asked that the papers not go to the residence that day.
We don't have to know what Cohen was doing or thinking. We don't even need his testimony. We have David Pecker, Hope Hicks, the accounts payable guy who prepared the checks, and Donald Trump's signature on the checks.
We have testimony from Trump's employees, past and present, that Trump is a complete control freak who knows every detail of every dollar he spends. They have hand written instructions and notes written by Trump's employees, and Trump's signature on every single check.