I never used stable coins but as far as I know, exchange charges btc as fee when you send btc, eth when you send eth and USDT when you send USDT. If that's the case, how can the fee turn into $10-$20. Looks like I have missed a little to some part of your theory.
Fee will increase when unconfirmed transaction increase and btc/eth value increased. It happened last crypto boom(2017 dec). and you can see another up movement last week.
Here is the link for block details