The new BA website looks like it has been programmed with the probably well-meaning intention of keeping everything simple and catering smoothly to requests. Unfortunately many trips many of us call simple are not so to the new system. When I ask for business class I don’t necessarily mind having the main flight in business and the feeder flight in Economy but only when I am told so before I make my decision to book.

I have replicated the experience described by the OP and this information is most definitely obfuscated. You can see if you click on individual flight numbers exactly which cabin has been selected but you have to know to do so. In my opinion this is definitely misleading the consumer.

Indeed business class was available for the combined flights I selected but only in a higher booking class than was available for the first sector. In my opinion the new BA website was not fulfilling the search criteria correctly or even giving the consumer the option to do so.