That is an unwarranted assumption. When taking a plea the defendant, under oath, doesn't get to just randomly spew whatever crosses their mind. The prosecution and the judge are responsible, in accepting the plea, for verifying that the defendant is being truthful. In regards to that, and given that Cohen plead out in the first place, there has got to be a good bit of evidence supporting his plea...and I'd say that it is very likely that some of it is even more blatantly incriminating.
Cohen's lawyer released a tape to the press of Trump and Cohen discussing the logistics of making the payoffs. The feds seized Cohen's papers, and since this was apparently a crime there is no client privilege applying to conversations in which Cohen and Trump conspired to commit it. The reality here is almost certainly that Cohen didn't get a cooperation agreement because the prosecution already has so much blatantly incriminating evidence against Trump that his testimony would be totally superfluous.