One of the main theses of Harry Potter is that blood counts for nothing. So Hermione is born with magical powers despite her non-magical parentage, and some children in Wizard families ("Squibs") are not born with magical powers despite their magical parentage.
The point that remains is that all the powers of Wizards in Rowling's work are indeed innate, no less so than Galadriel's or Gandalf's; but, like Galadriel or Gandalf, the children in Harry Potter require instruction to fulfill their magical potential. Galadriel studied under Melian, for example.
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Among the friendly dead, being bad at games did not seem to matter
-Il Lupo Fenriso
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