I've just been onto the birth registration of children born abroad pages of the gov website, and it is still as I thought.
Nobody needs to register their children with the consulates. All you are doing is spending nigh on £200 for a bit of paper.
They are really naughty IMO about this, because while they state "you don't have to do this" they also say "this may make it easier for you in future" which is very vague. In what way? I know my child is British, she travels on a British passport, which the ppt office wouldn't have given her had they not checked she was British through me!
The confusion often stems from two things a) the word "registration" used for both these "birth certificates" and for certain sections of the BNA where you have an entitlement to British C/ship through, for example, grandparents, so you apply for "registration". Often the consulates let you think you need to do a) in order to have BC/ship by descent. The other thing is the consulate staff deal with nationality as a bit of a sideline together with their other duties. They work for the Foreign Office, not the Home Office, so don't have the in-depth nationality training.
In fact a couple of years ago, on here, I helped a couple of families who realised they had paid for registration of their children (as BCs, not birth regn) when it had never been necessary.