It seems to me that the cost of transactions will increase along with the increase in the value of coins and this applies not only to Ethereum. The same situation can be observed on the Tron platform, when the cost of transactions has risen from one Tron to 10 Tron. I am sure that the situation will be the same with regard to BSC. Nevertheless, today it is very profitable to transfer Ethereum using BEP20.
True, in general the value of the Gas fee will always have an effect on the unit value of the token itself, because the expensive or the low cost will be seen in the token price in the market, so everything will be almost the same as Ethereum.