When Ginny joined the Gryffindor Quidditch team and was showing talent, it came as a complete surprise for her brothers. George said:
“Actually, I dunno how she got so good, seeing how we never let her play with us. . . .” “She’s been breaking into your broom shed in the garden since the age of six and taking each of your brooms out in turn when you weren’t looking,” said Hermione from behind her tottering pile of Ancient Rune books. “Oh,” said George, looking mildly impressed. “Well — that’d explain it.” ~Harry Potter and the Order of the Phoenix, Chapter 26: Seen and Unforeseen
Could be that her family was protective of Ginny, considering her their little baby girl, but I don't recall even Mrs. Weasley forbidding Ginny to participate in any sports activity. She mainly opposed Ginny being told Order secrets related to the return of Voldemort, which was understandable. And even in that case, Ginny's brothers had no problem with Ginny getting information Mrs. Weasley considered too dangerous and thus unsuitable for her age, for example, Sirius' and Lupin's update on the Voldemort situation at the beginning of Harry's fifth year:
“Asleep, yeah, right,” said Fred in an undertone, after Hermione bade them good night and they were climbing to the next floor. “If Ginny’s not lying awake waiting for Hermione to tell her everything they said downstairs, then I’m a flobberworm. . . .” ~Harry Potter and the Order of the Phoenix, Chapter 6: The Noble and Most Ancient House of Black
So if they were fine with Ginny being told Order secrets, why not let her join their Quidditch games? After all, they let Ron join and from what I could tell they were also fine with girls (Angelina Johnson, Alicia Spinnet and Katie Bell) playing Quidditch on the Gryffindor team, so why not play with their little sis?
Is there something in the Weasley family dynamics I am missing?