I'm not sure that has to do with privacy and more like "help us avoid paying income taxes." No one denies that cash is completely anonymous, but that's not the intent here. This is coming from a huge advocate for privacy, but to me this is a stretch.
Avoid paying tax and avoid paying the card transaction fees, which I think is usually a few percent. The latter is valid to get a discount, but it's not going to save them 10%, so feels a bit dodgy.
Or just raise your prices instead of hitting customers with surprise fees. I own a small business. The vast majority of our revenue is from credit cards. We don't pass those fees into customers, we just raise our prices as needed.
This is because the system hates cash and has erected all types barriers and inefficiencies when handling cash.
You should look into Monero, it is like cash but digital. No more worrying about your payment processor cutting you off because of you political views.
According to my friendly neighbourhood corner shop owner they pay 3% on card transactions, unless it's over a certain amount and then they pay a flat rate.
But he's not sure what that amount is because as a tiny corner shop they basically never hit it.
Perhaps you'd pay a flat rate if you were paying for a PS5 or something in that price range, I'm not sure.
Credit cards transaction may only cost a few percent, but the cost to business can be significantly higher. For example visa forces you to no require a minimum for credit cards, and they charge a base fee regardless. Also if a customer issues a dispute, the business may have to pay the dispute fee, regardless of the outcome of the dispute. Louis Rosmann as quite a few good videos on this topic, eg: https://www.youtube.com/watch?v=6an5ZP4Vnzg