It definitely follows the structure of the original VERY closely, it's just different storytelling. I think it's a fair criticism. I happened to see Into the Woods the other day for the time since its release, and I completely forgot how stagey the second half is (not in a good way). MPR is more cinematically justified, I'd say, but I can understand your perspective there completely.
I couldn't identify the problem elements at the moment, I'd have to see it again. But pretty much all of her songs (or participation in songs) were pushing the Openness vs Preconception argument really strongly:
Can You Imagine That? -- "So, perhaps we've learned when day is done/Some stuff and nonsense could be fun/Can you imagine that?"
A Cover is Not the Book -- "Chapter titles are like signs/And if you read between the lines/You'll find your first impression was mistook/For a cover is nice/But a cover is not the book"
The Place Where Lost Things Go -- "Nothing's gone forever/Only out of place"
Turning Turtle -- "When you change the view from where you stood/The things you view will change for good"
And in each one, she kind of influences the kids (and Topsy in Turning Turtle) to open their minds to whatever she's said (they suddenly get very excitable after CYIT; they start seeing Wilkins as a wolf; they think of their mother as 'gone but not forgotten'; and Topsy decides she loves Wednesdays because she's 'quite contrary'). The OS Mary is the one that's just kind of leading the kids through these adventures, but IC Mary is the one that has the 'lessons'.
I really, really believe this is an MC handoff. Because you have that really strong MC scene with Michael where he's speak-singing to his wife about how he misses her, and when Poppins arrives, it's the kids that pick up the 'I miss our mother'. They're all so consumed by their grief that they've (in the words of Dame Angela Lansbury) "forgotten what it's like to be a child!" Nobody else deals with that in this story outside of those four.
It's possible I'm filling in blanks. But I do think they spelled out an argument of sorts -- I just don't know how much is there until I see it again.