How about a study backing it up? https://www.thelancet.com/journals/laninf/article/PIIS1473-3099(21)00648-4/fulltext
Interpretation
Vaccination reduces the risk of delta variant infection and accelerates viral clearance. Nonetheless, fully vaccinated individuals with breakthrough infections have peak viral load similar to unvaccinated cases and can efficiently transmit infection in household settings, including to fully vaccinated contacts. Host–virus interactions early in infection may shape the entire viral trajectory.
I don’t know if you’re playing dumb or are actually not understanding this but I’ll lay it out for you anyway.
A) If someone doesn’t carry the virus they can’t spread it.
B) Vaccinated people are much less likely to carry the virus than unvaccinated people.
Ergo, vaccinated people are less likely to transmit virus than unvaccinated people.
Additionally, in the very study you are quoting which claims peak viral load is similar in both the vaccinated and the unvaccinated there is also evidence that viral load decreases faster in vaccinated patients.
This is aside from the fact that it has always been assumed (with at least one
study supporting the idea) that vaccinated people with COVID are less likely to transmit virus than unvaccinated people with COVID. You’ll need more than one study to confidently state that vaccinated people transmit virus at the same rate as unvaccinated people.