>they pick the rates. So they can set those rates to generate exactly the same profit for them as with fees.

No, the rate is set by the card network, eg https://usa.visa.com/support/consumer/travel-support/exchang...

Huh. I wonder how long that's been true. I guess I haven't been to America since last century, but I could have sworn I had zero fee but unreasonable FX rates when I was in Poland which can't have been more than 25 years ago.