No, the offense was not just lying. He willfully falsified business records (fraud) to cover up another crime (campaign finance violations) to dupe the public and influence the 2016 election.
I'm not sure your claim is accurate. If he had used campaign funds to pay Cohen back, that would have been an even clearer campaign finance violation and attempt to interfere in the 2016 election.
No, they had to come up with a way to do it outside of the campaign. What better way than running it through your business where Cohen already did work for you and it could be more easily explained away.
The point was he caught and killed the affair stories because they were politically damaging and the Access Hollywood tape had already damaged him a lot. He developed that strategy, along with publishing false hit jobs on his opponents, with Pecker & Cohen.
Then, he had to pay Cohen back because Cohen refused to do his dirty work for free. He instructed Weisselberg to create fake work entries and checks (fraud) to pay Cohen back in batches so it looked like a normal legal retainer agreement. They doubled the amount so Cohen would not lose half of it to taxes and also added in a $60,000 bonus for a job well done.
The evidence was overwhelming and on paper plus many witnesses who corroborated not only the arrangement, but the facts that Trump always checked every payment, signed most of the checks (in the White House when he was supposed to divested of his business) and was a micro manager that knew everything that was going on.
Trump had his business in NY and thus he was convicted there. However, I think this trial could have been in many other states and a jury of normal Americans would have found him guilty as well. Why? Because he is guilty and he knows it.