There are some trust issues that are still problematic like for example someone needs to provide a price feed against which the value will be stabilized. Whoever is in control of that price feed could of course damage the system to the point where people loose all their money.
But in the real world such a price feed could come from a DEX so its not controlled by someone but instead by what people buy and sell. In a closed system this could however cause a "runaway" of the price due to price feedback loops.
Here is an interesting talk about a crypt collateralized stablecoin proposal https://www.youtube.com/watch?v=Se2CDsmMqvE again this does NOT yet exist!
Traditional currencies are mostly backed by things that are very unlikely to crash like real estate, profit share in companies and other ressources with value besides speculation.
What would happen if BTC crashed to 1/10 of its price, and all other coins followed?
If you had a coin backed by a basket of other cryptocurrencies, it would likely be relatively stable in that the distinct individual volatility of each of the backing coins would be mitigated. OTOH, the overall volatility of the cryptocurrency market (which often moves rapidly and in the same direction) would not, so it wouldn't be very stable.
The key to a stablecoin however, is trust. Users have to trust that their holdings are actually backed by real fiat currency as that's the promise of the stablecoin.
That trust can typically be bolstered by things like 3rd party audits, which is something Tether promised for many years and did not deliver.
With Tether the problems are that they have avoided scrutiny and audit, made statements on their website that subsequently turned out to be false and have a somewhat murky corporate structure. None of these, should , engender trust.