For me, the biggest problem with the show is the whole "Voldemort had a child" thing at the center. The books make it very clear that in his quest to achieve immortality, Voldemort never believed the traditional notion that having a child carry on your legacy is a valid pursuit because he was so ashamed of his own parentage. And then the amount of coverup--Bellatrix would have been pregnant during the entirety of the sixth book, and the fact that nobody noticed or mentioned that just makes this whole thing feel like an afterthought.
Then there's the issue of the time-turners. Where Prisoner of Azkaban makes such careful choices in how it approaches time travel so as not to upset any rules, this one throws all that careful planning out the window and presents a cliche, pulpy genre story. Rowling said that she destroyed all of the time turners in the battle at the Department of Mysteries so that time travel wouldn't be a tool that could be used later on. Cursed Child's quick explanation of "new technological advances" shows sloppy world building and opens a much bigger can of worms than the one it cares to explore. The careful plotting and meticulous world building are huge parts of what makes the books so remarkable, and these qualities are lacking in Cursed Child.
And then there's Albus's journey... in the epilogue of Deathly Hallows (also the first scene of Cursed Child), Harry reassures Albus that if he is really afraid of being in Slytherin, he can just choose to be a part of Gryffindor, like Harry did. This moment reinforces one of the books' central themes, "It is our choices, Harry, that show what we truly are, far more than our abilities." The concept of free will always persists, and Voldemort's adherence to a prophecy is what initially causes his downfall. When Albus is sorted into Slytherin despite his wishes not to, Cursed Child ruins the emotional impact of the final moment of a several-thousand-page series.