In practice, I don’t know of anyone offering 0.3% card processing fees. Most POS systems and online payment processors are more like ~2% e.g. Stripe is 1.4% + ~€0.30 to take online payments from within the EU. And chargebacks are passed on to the merchant with an additional fee tacked on.

Can you point me to any cheaper solutions?

I like the idea of having the processing fees displayed prominently on the receipt though (like VAT). If people had to actually pay more to use Amex (rather than the merchant absorbing the blow, or distributing the costs across their other customers), then they might quickly go out of business (and rightly so).