Because that would be illegal market manipulation.

Any institute, including GameStop themselves, would be committing a crime if they knowingly triggered a short squeeze. The SEC rules on this is quite clear.

What's interesting to me is that the official SEC report on GameStop "squeeze" specifically called out that the rise in price WAS NOT due to short covering, but rather a large increase in retail buying and market makers balancing options. So the reported % short went from more than 100% to less than %20 with zero short covering and prices falling? Yeah, sure.