Netting & Lack of Pollination
We usually recommend the 7mm butterfly netting for brassicas, this keeps out the cabbage whites and caterpillars. With this netting you would have been able to grow your broad beans in the cage too, although they are not usually grown under netting.
The problem you had was definitely lack of pollination. I’m assuming you’ve purchased the insect mesh netting in which case the pollinating insects would not have been able to get through? You had flowers but they probably hadn’t been pollinated before they were moved into the cage. We did have a very cool and wet spring plus there is a shortage of bees so put all those factors together and you get lack of pollination. I would try to encourage bees into your garden by planting some bee friendly shrubs or some companion planting and leave your broad beans uncovered next year.
We have a Video on our netting page explaining the different types of netting we sell and how to use it.
I hope this had helped to answer your question.