> I've seen reports that say that if you have Covid, you transmit it as easily as anyone else, regardless of vaccination status.
While you're indeed infected, yes. If that duration for vaccinated persons is shorter than the duration for unvaccinated persons, then the obvious deduction would be that the shortened duration results in a lowered overall chance of spread - even if (indeed, especially if) the viral loads are equal (as vaccine-skeptics like to assert as if it's some sort of "gotcha"). Transmission is a function of viral load × time, so shorter time = shorter window of transmission = less risk of transmission.
Factor in the reduced chance of being infected in the first place (as you point out), and it becomes obvious on both fronts that vaccination reduces the spread of COVID, Omicron included.
That is why we see so many people saying that the vaccines don't prevent you from spreading it.
However, if you're vaccinated, you're less likely to get it in the first place, so from a standing start, you're less likely to transmit it. And since at some point everyone was covid-free, this is the important scenario.
Nobody who knows they have the virus should knowingly go near another person except in an emergency. Nothing you do will protect that other person.