Baked beans in tomato sauce are pretty well the only way dried beans enter British culture, as you say. They have been enormously popular over the last few decades. Traditionally marketed by Heinz. They are canned in tomato sauce. For the average British person the beans never appear without having been poured out of a tin (can) and never without the tomato sauce they come canned in.
I think it is probably possible to find dried beans in larger supermarkets but they would be a minority purchase.
Well, not to get ethnic, but with the large number of people in the UK of Indian/Pakistani origin (who are big legume eaters) as well as Caribbean (not quite as big, but they have their pigeon peas and such.) there'd be a lot of importing of dried legumes for those markets.
I think a big portion of the lack may be classist. For a lot of European history, dried beans were considered a staple, but a staple of the peasants. They were what you ate when your weren't rich enough to be able to afford significant amounts of meat. If you were a noble, you probably wouldn't be eating a lot of beans (except possibly on fast days, and even then it'd be iffy.) since you could afford better.
Add on the fact that, until the common bean arrived in the 1500's the base bean for much of Europe would be the broad* (or fava) bean. which meant that, in some areas, there was a real chance of local beans being poisonous to you (i.e. due to favism). The advent of the common bean, which was both a bit safer (you still needed to cook them right, but once you did, they were safe) and rather more productive in many cases. would have eased this a little.
But come industrialization, as people's affluence went up a bit and the supply/price of meat and other such luxuries became more affordable, the number of people who needed to rely on dried beans got lower.
Even here in the U.S. dried beans were usually considered "poor man's food", same as maize. The richer you were, the less of them you'd eat. Being raised to elite status again is a pretty recent thing.
Green beans stayed acceptable because they WERE a scarce commodity (having a short season) and demonstrated conspicuous consumption (you were so rich you afford to pick your beans early when they were still a vegetable, rather than wait until they matured, dried, and became more nutritious.) Same story as peas.
*Which is why I maintain that Jack's beanstalk would have to be a fava bean; it'd be the only kind at that time (plus it actually CAN free stand as a stalk, which a common bean can't).