The argument that vaccines don't prevent delta tranmission are based on headlines, not science.
And unfortunately right now science is playing catch-up.
There simply isn't any evidence on breakthrough transmission with delta testing humans yet because it is recent. The headlines and public health messages are assuming worst case in order to scare vaccinated people into masking up again, while there's actually indications that vaccinated breakthrough infections almost certainly transmit less.
What we know is that while early in the infection viral loads in delta are much higher, but Alpha, delta, and non-VOI/VOC infections feature similar viral trajectories[1], in vaccinated breakthrough infections RNA load declines faster[2] and the ability to culture virus in breakthrough infections is lower[3], we know that vaccines reduced transmission with alpha[4] and apart from scary headlines we still expect that vaccines reduce transmission with delta. While there's documented cases of delta breakthrough infections transmitting to one or two other people, there's not been any studies of how common that is, or how common it is for breakthrough infections to become superspreaders to >10 people, or if that is even possible. Since viral loads are correlated with transmissibility and symptomology/virulence the very high protection against severe disease with vaccination may also protect against severe transmission.
[1] https://www.medrxiv.org/content/10.1101/2021.02.16.21251535v...
[2] https://www.medrxiv.org/content/10.1101/2021.07.28.21261295v...
[3] https://www.medrxiv.org/content/10.1101/2021.08.20.21262158v...
[4] https://www.timesofisrael.com/80-of-vaccinated-covid-carrier...