Could some of the other funds have done what Reddit did with Gamestop?

When the music stops, whichever Redditors are left holding GameStop shares at that point will lose their shirts, because the shares are really only worth a few dollars. No serious fund is going to take that risk.

Short squeezes and gamma squeezes (the option counterpart) are not unprecedented. They are however quite risky.

An example of this happening before is the Volkswagen Short Squeeze.

JP Morgan squeezed the aluminium market back around 2015.

Here's Goldman squeezing aluminium back in 2011.

To some extent, the big players are defined by always looking for opportunities to do this. What they don't do, because it's effectively illegal, is go round publicising their holdings and trying to rope in others to the squeeze.

The people doing it care more about making an example of the short sellers, than they care about not each losing a little money.

A very different motivation than a fund manager.

Of course they could have. In fact, they are probably already participating, and the price movement is more due to the funds than to individual investors on Reddit. Just check out GME's statistics.

As of time of writing:

95% institutional ownership. Unless the fund managers themselves are on Reddit, the Redditors account for no more than 5% of GME's outstanding shares, and GME's price movements are still largely due to the funds.