Doing it on a card doesn't feel the same
I don't understand that mindset. You're still spending money and the card/cash is just the payment mechanism.
The OPs parents could use a credit card for all their spending and then just pay the bill off each month by DD, which overcomes their fear of a contactless card linked to their bank account. But I don't understand why people spend from their bills account, that must be a nightmare to keep up with.
There are different systems for the retailers and the market/charging structure has changed a lot in recent years. Some smaller retailers probably need to review what they are using, so they don't get hit with high charges which wipe out their profit on smaller transactions.
It used to be something like 30 p for a debit card transaction but 2% with no minimum for a credit card transaction, which meant that debit card transactions on small purchases didn't stack up for the retailer but credit cards were fine, but obviously the retailer can't be expected to interrogate every customer to see whether they are using a credit or a debit card, and most people seem to use debit cards, despite credit cards offering a lot of advantages in terms of cashback, purchase protection, cashflow management and being cheaper if the alternative is going into overdraft. But many now are a flat percentage for all transactions, which is usually more favourable to the retailer, especially those who's typical spend is low, eg convenience stores, takeaways, cafes. Of course handling cash also costs, but many small retailers will use a lot of the cash they take to pay suppliers and for their own personal spending, so they won't be banking all the cash that they take, which is not the same as not declaring it all btw.
But the system is not always 100% reliable and failure could cost the retailer a lot of lost trade if most of their customers have usually used cards and stopped carrying cash. On the individual's side, it's best to have at least two payment methods available, in case one bank's systems go down so you can use the other. It's probably also worth carrying a small amount of cash even if it takes months to use it, for the odd occasion where the system is down, or a retailer doesn't take cards or charges extra to do so. Eg, the parking machines charge an extra 25 p to pay by card, which is totally ridiculous, but lots of extra charges like that can add up.