If they sold a call option which is exercised, they’d HAVE to buy

Only if insolvency makes a sound. If the shorts don't have the trillion dollars to buy that stock to pay back the debt by the deadline then they are sol.

What deadline exists for shorts, assuming these aren't puts?

Shorting doesn't only mean short stock. You can be short call options, and if they are in the money, the holder of the contract can exercise his or her option, which forces the seller of the contract to provide shares at whatever price.

For every long call option contract, there is a counterparty who is subject to unlimited upside price risk, just as a traditional stock shorter. The difference is the contract has an end date.

I would argue that options account for more of the pressure/volatility of these recent stock prices than the shares themselves.

Today was the monthly option expiration day.

But of course not all options expire at the same time