The OP misses the point. The purpose of putting Cohen on the stand is to provide headlines for the Leftist Media to badmouth Trump. It doesn't matter whether they are true or not; nobody will pay attention to the cross examination or the proof that Cohen is a liar and a scoundrel.
More importantly, this trial involved no crime, and everyone with a three digit IQ already knows all of the relevant facts, although two significant ones are un-knowable: whether Trump actually fucked her, and Trump's motivation for trying to keep her quiet.
There are two purposes for this trial: (a) to embarrass Trump, and (b) to be able to go through the Summer General Election campaign calling Trump a "convicted felon"...even though it is a certainty that this conviction will be tossed on appeal. I would even suggest that the DA and judge might be sanctioned by the NY Bar Association for abuse of discretion and bringing infamy to the legal profession. But that's just me. (I'm an attorney, and member of the Pennsylvania Bar).