I'd imagine it's because football is (rightly or wrongly) encouraged pretty much everywhere (for boys and girls).
The rules are clearer and there will usually be someone around who can monitor how it's being played and how safely it's being played.
The issue with gymnastics is that the children aren't necessarily mindful of who is around them before they launch themselves into the air so some injuries happen that way. This is especially the case if they are effectively doing a 'routine'. And there likely to be fewer people (if anyone) around who are knowledgeable enough to recognise if a child is competent.
And, don't forget, we act in loco parents we are not the same as parents. What parents are happy to allow their children to do under their supervision is not the same as what we can allow them to do when under ours.
She's not suggesting that all gymnastics is banned just for clarity around whether aerial type moves are allowed. Because if she knowingly allows them to do something that the school doesn't technically permit then she will be held responsible for any injuries that do occur because the HT will just say, "Well, you shouldn't have allowed them to do that. You know we don't allow that."
And, tbh, if a parent posted on here after their child became injured doing a 'technically not permitted" gymnastics move at school (eg an aerial), the majority of posters would not be saying, "Blimey, OP, whats your problem? My childhood was spent doing cartwheels on the school field!"
Would they?