Intent aside, they still have to prove that "Trump decided to make or cause the false entry in the business records" which is not a given as your assertion implies.

Of course. They have to prove every element of the crime.

But this isn't some subtle, mysterious, ineffable theory of criminality that legal minds are puzzling over. The prosecution's theory of the case is pretty straightforward. Trump paid two women not to disclose that he had affairs with them so that it wouldn't hurt his chances of getting elected. Therefore these payments constituted an in-kind contribution to his 2016 federal election campaign. In order to make sure that these payments were concealed, he deliberately structured them as if they were payments for something else, which caused them to be falsely recorded in the business records of the Trump Organization. Because these payments were illegal campaign contributions, the false records were intended to conceal another crime - and therefore constituted a felony.

Does the prosecution have good evidence to prove their case? We'll find out. It's indisputable that one of their key witnesses has serious credibility issues, and it's never easy to prove up a specific intent crime. The jury might not find that the evidence shows the payments to be a campaign contribution. They might not find Cohen credible. They might think that the prosecution lacked evidence of Trump's state of mind at the time the records were created. Etc.

But we're long past the time when anyone should wonder aloud what object crimes the prosecution is alleging took place, or what evidence they have to support those allegations. They made those arguments to the judge months ago, and it's not hard to find them.