I'm really not sure if we're on the same page on the conclusion of the study, so I'll repeat it just in case:
Scenario 1: A vaccinated household member (the index case) is COVID-positive. The study finds that other household members are just as likely to get infected regardless of the vaccination status of the index case. There is no reduction of spread.
Scenario 2: Some household member (regardless of vaccination status) is COVID-positive. The study finds that other vaccinated household members are less likely to get infected. Spread is reduced.
That's basically what the WHO is saying - we may have had the positive impact of scenario 1 prior to Delta, but that's mostly gone. Scenario 2 still shows some effect on spread. It may be lower now, but it's still significant.
The article also states that "Vaccinating school-aged children may help minimize school disruptions by reducing the number of infections at school" and "there are benefits of vaccinating children and adolescents that go beyond the direct health benefits. Vaccination that decreases COVID transmission in this age group may reduce transmission from children and adolescents to older adults".