That's the extent of my spoiler free review, no promises from here on out.
I'm not sure what everyone expected from the story, the most common critique I've heard seems to be that it recycles Voldemort as the old villain. But were Rowling and her co-authors supposed to introduce, develop, and then defeat an entirely new villain in the course of four acts? Not to mention that a Harry Potter story without his nemesis in some form or fashion would seem . . . anticlimactic.
I loved the depiction of our older heroes--not unscarred by their earlier adventures, but absolutely building their lives and as happy as anyone can expect to be. Despite a bit of slash fic wariness at first, I loved Albus and Scorpius as the new heroes; they were absolutely adorable. The format of reading as a stage script instead of a novel was less unsettling than I expected, although I'm fairly certain that the stage directions were embellished for the reading public.
Yes, it fails in many respects. There's plenty of cringe-worthy moments,
I didn't love the time travel in principle because it's a cheap and messy plot trick, but I loved the exploration of the alternate histories that it allowed. There's a reason it's such a popular cheap plot trick. Specifically the scenes with Snape; I lost it a full half page before Scorpius' revelation of the existence of one Albus Severus Potter, I could just see it coming.
But on the whole I didn't feel disappointed, possibly because I had low expectations? There were many moments and quotes that I loved:
ROSE: The rumor is that he's Voldemort's son, Albus
A horrible, uncomfortable silence.
It's probably rubbish. I mean . . . look, you've got a nose.
Slightly ashamed to admit that I laughed. Suspect I have a terrible sense of humor.
And now before I hit publish, I have to mention a point that the brilliant Haley at Carrots for Michaelmas raised in her review. (I was slightly terrified to read said review because she didn't like it and I was afraid that I'd be convinced. Her literary analyses are that good, but I escaped with warm feelings intact this time.)
Huge plot flaw: Voldemort as Delphi's father. Voldemort as anyone's father. Lets skip the thinking about Voldy's sex life (ew) and think about the whole generating life thing. By Deathly Hallows, Voldemort has shredded his soul beyond recognition, been killed once, and undergone some serious physical changes. Death does not beget life; participating in the creation of a child should have been far beyond him at that point. Now obviously its Rowling's world, but it just doesn't seem to fit with the themes of death and love that run so deeply and beautifully throughout the rest of the series.