JustKia
As you say, the birth names only are given in the GRO index, so if a child doesn't know he was adopted, or even if he does, but does not know his birth name, he won't be able to find it without applying to Kew and receiving counselling. This is what happened to my adopted brother, born 1936.
My own children, adopted in the 70s, are more fortunate. We were given full details of their natural parents by the adoption society, right down to physical appearance, school achievements, etc. It's intriguing to look at what they have inherited - good at sport, for example as well as curly hair and eye colouring.