It seems to me not correct that the price of most ERC20 tokens depends on the price of the ether! What do you think?
Most do yes, but not all. Those that have an immediate dividend possibility are clearly in demand and they do preserve the value in terms of fiat.
Also, I have observed that coins related to crypto exchanges frequently preserve their value quite well because these are value backed by exchange´s fees, so their value is good even when other tokens fall because at the end of the day you will normally sell coins and pay a fee for it and it does not matter if the market is good or bad. In that category you will find Binance, OK and similar coins.