Yes I agree with you. I wish I could find a study that clearly shows that unvaccinated people are more likely to transmit the disease than vaccinated people.
There are studies, but the ones I have found compare only people are infected. And when you have COVID you're contagious, vaccinated or not. This is then used as a basis for claiming: "See vaccinated are just as, or even more, contagious as unvaccinated!". That's what I'm trying to disprove.
Example of what I have found: https://www.thelancet.com/journals/laninf/article/PIIS1473-3...