Don't read any of the nonsense circulating in the media. There is a reason we don't try criminal cases by Tiktok, by superficial appearances, by bias - unconscious or otherwise - or by journalism after the fact.
The New York Times is, by reputation, a high-quality publication. But journalistic pieces are by definition opinion pieces. They're not, and are not intended to be, fully evidenced-based analyses, they don't rely on or cite bona fide academic sources, they are not written by experts in the field they refer to nor peer-reviewed by a panel of similar experts. On these bases alone the author of that article has a lot to answer for, IMO.
For a more informative summary which takes into account the letter of the law and the admissibility or otherwise of the evidence presented at trial, read the appeal judgement. It's in excess of 50-pages long but it's written in accessible language and gives detailed reasoning for the rejection of an appeal on the basis of all objections put forward by the defence.
Justice is not infallible, and miscarriages of justice are not unheard of, but as a justice system this is the best we have. It's far preferable to knee-jerk emotional responses (remember Louise Woodward, another cherubic-faced nanny figure also guilty of infant homicide?), vigilantism, or trial by media. At least you can follow the legal reasoning as to why the appeal was rejected.
If you're going to listen to any media, the 'Double Jeopardy' podcast by Tim Owen and Ken Macdonald is as good as any. At least they are experienced KCs - and in the case of Owen, a long-renowned expert on miscarriages of justice which includes insight into Sally Clark amongst others. Owen discusses the reasons why defence counsel routinely take the decision not to include expert witnesses, has also read the appeal judgement, and sees no irregularities in that document or the way in which the letter of the law has been applied.
His conclusion is also that it's fine to be sceptical and question the legal system, but you have to be able to show exactly why it's wrong (a thing the appeal application singularly failed to do). You can't just appeal against a conviction because you don't like the outcome. A lot of what's out there on the www is simply wind and hot air.