I qualify. Are you asserting that the HOA case will be held in Family Court?
No, don't bother. It's obvious that you have strongly held, but erroneous beliefs. There's nothing gained by me refuting you point by point, because you're obviously here to make a point, not learn.
Children cannot be a party to a contract, because minors can't be held accountable for a contract. Any contract signed by a minor is null and void on it's face. They cannot, therefore, have standing as a party to a contract. The child is asserting no right to live there, the GP are resisting being compelled to uphold a contract they made.
Family law, on the other hand, carves out a legal sphere where "the interests of the child" are considered. The child is not a party to the marriage; the court considers all issues with the best interests of children in mind.
No, this case would not be heard in Family Court.
When a general type of contract is known to by its nature cause harms to society, that type of contract usually should be and is declared void. Such is the case with old real estate titles that declare that "in perpetuity this parcel may not be sold to negros or jews" or such clauses. Those clauses are today considered void.
This HOA has a contract which harms the children as a class of people in society, families as a class, and in this case harms a particular innocent little three year old.