Cursed Child sounded like ridiculous, over-the-top fanfiction, so I didn't read it and personally don't consider it canon, Word of Rowling or not.
However! As far as the books are concerned, you can only choose your House if there's a legitimate conflict between which House you should be in; this is what's known as a "Hatstall" (found in the Sorting Hat notes from Pottermore, which I do consider canon because most of it is based on notes that Rowling wrote as she was writing the books; according to the site, McGonagall and Flitwick were both Hatstalls for Gryffindor and Ravenclaw). Remember, Harry never asked specifically to be put in Gryffindor; he asked for not Slytherin. Whatever the Hat chose from there was entirely a toss-up between three other Houses, and it immediately Sorted him into Gryffindor. But the Hat's word is final; according to Pottermore, Neville spent about five minutes arguing with the Hat that he should be in Hufflepuff, but the Hat was resolute and finally put him in Gryffindor. Meanwhile, characters like Malfoy were Sorted the minute the Hat touched their heads; they didn't have time to make any choices, whether they would have done so or not.
And Sorting isn't a cut-and-dry thing based on just your personality, either. People tend to simplify the Houses into "brave, nice, smart, and evil" categories, which is where all the whining about "why wasn't Hermione a Ravenclaaaaawwww" comes from. But Sorting doesn't just reflect basic aspects of your personality; it also reflects what you value and how you react to the world. To take the Hermione example: she doesn't just read a lot and get good grades; she's highly competitive, wants to be the best, vaunts her opinion over everyone else's, and shouts down people who disagree. She's bossy, fiery, and passionate, she seeks justice for the oppressed (sometimes deciding what they need instead of listening to them), and yes, she's incredibly brave. Hermione is absolutely a Gryffindor; it's just that her academic prowess and know-it-all behavior tend to override all of those other traits in people's minds because they insist on categorizing her into this flat "smart girl" archetype.
it's not about 'deserve'. it's about what you believe. and i believe in love