You are right. It's unlikely to be open ended, although open repos do exist. It would usually be possible to roll the loan agreement too, with the payment of an ongoing fee. In the case of Gamestock, if any institutions were holding, they'd probably want their stock returned so that they could cash in on the massive gains!
However, as you know the problem with GME is that apparently the number of shorts outnumbered the actual stocks in circulation, and now that the squeeze is on, it's driving it sky high. It is mad though, because the underlying company probably is a basket case, hence why some hedge funds were so heavily short.
I still don't think this will end well for most of the private investors. Quite a few have said they don't care about the money though, and are happy to have taken on Wall Street a bit, and caused a stir! One good thing to come out of it might be to make hedge funds / institutions to think twice about shorting companies quite so hard in the future.
Incidentally, the guy who is being kind of credited with kicking it all off apparently had a $50,000 (long) investment in Gamestock dating back to 2019. If that's true, then given the company's realistic potential (having not adapted very well to the online world), he's super lucky to be walking away with a huge profit! - A bit like winning the lottery really - the fact that it went viral like it did.