You don't need an "emergency infusion of 3 billion dollars" if you just bought put options that are now deep OTM. As you mentioned, the total downside is capped at 100%.
They clearly had real exposure and serious panic. If they were using put options they were leveraged or had non-standard terms (not the same as typical retail investors)
"Melvin’s most recent filing showed that it held 5.4 million puts on GameStop, valued at more than $55 million — an increase of 58 percent during the third quarter."
If it were just puts, then their exposure to a run-up would have been 55 million, not billions. Puts can only go to zero. The filing merely hinted that they were shorting the stock because they also had some puts on it.
Think about it, if you buy a $9 put for $10, the most you could lose is $10 when the stock goes way above $9. The unlimited loss scenario with options is writing (selling) a naked call, where you receive a premium but if the stock price rises above the strike price your loss rises.
Melvin's position must have been mostly short equity. Perhaps only a few million shares of a $3 stock.
[1]: https://www.institutionalinvestor.com/article/b1q8swwwtgr7nt...