But HODLers don't really care about tether, do they? Isn't tether mainly used for traders who want to temporarily move out of crypto and use tether for that?

It's been suggested that a large percentage of the crypto market cap is based on tethers which have been used to prop up prices but also used to hedge against falling BTC prices. Essentially, bitfinex prints tether whenever the price of BTC drops, and in the meantime, people use these printed tethers to hedge against BTC (and buy back in when the price trends upward again).

So there are many iterations of compounded phantom value folded into the market cap of BTC. If tether is found insolvent, essentially 80% of the value of BTC disappears overnight.

The entire value of the crypto market depends on people's belief that one tether is worth 1USD. Or so the theory goes.

I've heard that theory, but I can't say I understand it. Let's say USDT goes to 0 instantly. All traders lose their parked money. I can see a crash from panic and from people recouping their lost safe money, but what else does it have to do with with cryptocurrencies?

If USDT would go to 0 instantly many things would happen (a few of the biggest BTC fiat markets would not be able to operate, such as binance's main BTC market). As a practical example big traders would obviously sell their tether straight into crypto on those markets. But since they don't want to be exposed to that much crypto they would sell the same amount on other exchanges (such as Coinbase). In these chaotic events the biggest traders will fly to safety for most on their books (actual fiat, or hedged via derivatives).

But that's not the potential issue being described: Tether is supposed to be worth 1 dollar. A lot of big crypto markets run on tether. If it turns out that a lot of crypto was bought with synthetic dollars (tether) that turned out to be a lot less than a dollar, that would mean the price of many crypto projects got to where it is by imaginary money. In another word: overpriced.

If you're running off a cliff and you know there's a safety net to protect you, it makes the act of running off a cliff safer. Ergo, more people will run off the cliff.

What happens when that safety net vanishes as hoards of people are running off the cliff?

Edited to add:

This isn't really a theoretical issue, either. Risk is part of the price of a security. Something that has a high risk, may be valued less by investors. Often other investments will carry the same amount of reward but less risk.