Why would the cost of mining not fall correspondingly? It will adjust until it becomes profitable again, nobody is going to run mining farms if they are clearly losing money every single day.
So, given the number of miners drops too far too fast, Bitcoin may end up in a situation in which the global mining power is too low to even reach the end of the adjustment window in a realistic timeframe, which in turn triggers even more miners to shut off their equipment and sell their coins, which lowers the price even more and lengthens the timeframe, which in turn...and so on. This situation is known in Bitcoin circles as the "chain death spiral", and it has been a purely theoretical thing - until now, at least.
This is definitely true. Luckily for miners there are two other blockchains their mining equipment can make use of which don't suffer from the slow difficulty adjustment. They are both version of Bitcoin Cash (BCH and BSV) and they both have 10-block adjustment periods. This makes them much more resilient to sustained periods of depreciation.
* Of course it wouldn't be a smooth decline so there may be short periods where it occasionally becomes viable.
Many of the biggest Bitcoin mining operations are in places like China, near hydro-electric dams, where the electricity cost is extremely low. Labor is also cheap. Since they've already paid the setup costs, they are not going to be shutting down even at $1000 BTC.
There have already been hash rate wars. BCH has tried sending BTC into a death spiral before, by luring over miners to their blockchain. It worked, for a bit. Eventually, transaction fees went up on BTC, and miners were incentivized to return to BTC.
The average block time is 10 minutes for Bitcoin. Hash rate adjusts every 2016 blocks. That's 14 days on average. If the hash rate dropped 50%, it's 28 days.