The days when Star Trek looked at shows like "Maury" and "Jerry" for inspiration. It's pretty garish, isn't it?
Why they couldn't relinquish her and her parents' powers the way they did with Q is a bigger plot hole. It might have been addressed but I found far more fun topics in the episode to mention instead:
Q: They had assumed human form in order to visit Earth, I suppose for amusement. But in vulgar human fashion they proceeded to conceive a child. And then like mawkish humans, they became attached to it. What is it about those squirming little infants that you find so appealing?
CRUSHER: I'm sure that's beyond your comprehension, Q.
Q: I desperately hope so.
TROI: What happened to Amanda's parents?
Q: They died in an accident. None of us knew whether she had inherited the capacities of the Q, but recently they've began to emerge, and as an expert in humanity, I was sent to investigate.
Like an abusive marriage where they left their baby in a dumpster, it's complex. Q was deliberately causing trouble to numerous species. Amanda wasn't and deserved a fair chance. Humans have condign punishment for those who commit frequent heinous crimes we fight to protect the innocent who haven't committed any. The Q just wanted to be parents. Like a father who didn't realize he had a kid just like him but now takes an interest. The episode is hazy on more precise details, though it allows some continuity with what VOY did with the Q, which explains why they'd choose an Earth ghost town then a civil war.
Of course, it's also pawned off:
PICARD: Well, the circumstance of their death's quite odd. A tornado somehow escaped the weather modification net and touched down in only one spot. Amanda's home.
Q: Well, you never can predict the weather.
PICARD: But tornadoes develop from existing storm fronts. But you see, there were no storm fronts in Kansas that day. Witnesses reported that the funnel materialised spontaneously directly over Amanda's home, destroyed it, and disappeared.
Q: If you say so. I wasn't there.
PICARD: Were Amanda's parents executed by the Q Continuum?
Q: And what if they were?
PICARD: Then I think she has a right to know that before she makes a choice about her future.
Q: Don't be foolish, Picard. She has no choice. She never did. If she's truly Q, then she must return to the Continuum where she belongs. But if she were some kind of hybrid, neither human nor Q, then (a gesture)
PICARD: You would be so despicable?
Q: Don't be naive. You have no idea what it means to be Q. With unlimited power comes responsibility. Do you think it is reasonable for us to allow omnipotent beings to roam free through the universe?
PICARD: So what have you concluded? Does she live or does she die?
Q: I haven't decided yet.
Q nails it with "With unlimited power comes responsibility.". He knew doing illegal drugs is bad therefore using means to keep the kiddies from using them might not be a bad thing. If kiddie says "Well you did it so it's okay" then tell the kid of all the neat things that could happen if they do (brain damage, organ failure, diseases, death, etc...) Oh, wait, he knew tormenting zoo critters is bad and they need to keep their powers a secret from the rest of the zoo (for "zoo" read "universe"). Like the Doctor escaping the Time Lords to visit other planets for cheap thrills.
Of course, Q discusses hybrid but also states both parents were Q and taking human form. If both parents have dominant Q-Factor genes, then so is the offspring. Couldn't they have been harmless hurricanes in the sky looking down from a distance? Why would they bother to take form? Copulation is overrated as far as sensations go. Were they renegades like our Q?
This is also the same Q that decided humanity must be quashed and letting Picard create the temporal anomaly that fizzled out Earth's goo, all while giving hints on how to save everything...
Q: Instruct here. If that child does not learn how to control her power, she may accidently destroy herself. Or all of you. Or perhaps your entire galaxy.
It's as simple as that, regardless of underlying cause(s). She had it, she needed a chance to learn in a proper and relevant environment. No human could. This is so far beyond the human experience.
Lastly, the episode also makes for a far, far,
far more accurate LGBT allegory than the episodes that were alleged to be such allegory officially (yes, the inane test tube doped up baby vs old fashioned random acts of copulation episode from season 5 called "The Outcast). Like Guinan told Geordi about his dating life in "Booby Trap", it helps to be natural and not trying to force it.
DATA: Captain, I am reading a massive energy fluctuation in the planet's atmosphere.
PICARD: On screen.
(the murky brown smog clears to reveal white clouds and blue oceans)
DATA: Atmospheric contaminants have dropped to less than one part per trillion. The ecosystem has been restored to its natural state.
PICARD: Amanda?
(Q reappears)
Q: I told you it would be harder to resist than you thought.
AMANDA: I couldn't let all those people die
(Amanda summons Crusher)
AMANDA: Ever since I got here, I've been fighting this. I've been denying the truth. Denying what I am. I am Q. Doctor Crusher, I've decided that I can't stay. I can't stay here.
Q: Well, now that you've come to your senses, let's go.
AMANDA: No. I want to go and see my parents first. It's going to take some time to explain all this, so you'll have to be patient.
(Amanda hugs Beverly)
AMANDA: I hope I can come back and see you.
CRUSHER: You're a Q. You can do anything you want.
Remember when Riker was given the power of the Q and he managed to not use the powers despite throwing a tantrum, and he wasn't a Q at all? (One by proxy or as a cheap trick.) Yet Amanda and family were 100% true Q and couldn't hold back? Oh wait, continuity check (though to be fair, lots of people wished season 1 never existed but whatever...) And there's a sequel or prequel or something just waiting to happen. I suppose.
