Come on, now you're just inventing issues.
If they can get the shops to support them with collecting donations, do you really think the store would deny them the use of their own WiFi or demand they send their own cleaners in to wipe over the terminals?
People who run burger vans and even Big Issue sellers sometimes have card readers nowadays - I'm sure these household charities could afford to invest a tiny amount for one in each store if necessary. Running charities and fundraising costs money - everybody knows that - and they'd cover the costs im no time.
in reality, the stores themselves would likely pay for a terminal - just like I'm sure Tesco doesn't charge the charities it supports for a contribution to buying the big boxes for customers to put the blue tokens into. It would be immensely easy and very cheap for them to install a charity terminal and then put up a sign next to it to tell people which charity it's for this week/month/year - probably cheaper and simpler for them than emptying out the blue tokens, counting them and then processing a payment manually.
If the charities don't want to benefit from the huge amount of potential donations that people would make by card, there's nothing stopping them from putting out an old-fashioned tin for cash donations only.
Your penultimate paragraph really does go to perfectly prove one of OP's main points: this bizarre assumption that, if somebody doesn't give to YOUR charity at the time that YOU ask, it must mean that the person never gives to any charity at any time.
This is one reason why so many people feel awkward saying No, even if they don't freely want to (or can't) give on that occasion. Many charities - and people collecting for them - seem to have the strange idea that they are the only charity out there; or at least the only one that matters.
I don't recall personally saying that it was a 'terrible intrusion'. I have no issue myself with declining to give, if I don't wish to, but there are a lot of people who worry about what others think of them and would feel guilted into it - especially if it's their local shop and they know the assistant well, albeit the assistant has no idea what other charitable giving or volunteering they may do.
Moreover, by making it a compulsory part of a transaction (whichever button you press), people with poor eyesight and/or learning difficulties may well not realise that it's not actually part of the transaction to pay for what they're buying, so although it might not be a large amount to most people, they're effectively being laid open to being 'tricked' into giving.