"So much spread" is relative, in this case relative to how much spread you would have had in an identical unvaccinated population.
When we have tested these vaccines in a controlled way we find that relative to unvaccinated people there is a lesser chance of symptoms, that disease is more mild, and that it doesn't last as long. That's what those phase 3 trials and subsequent studies point to, and that bears out in general if you look at the number of people being hospitalized between vaccinated and unvaccinated.
All of these things align with lower amounts of viruses in your body (which also aligns with transmission potential).
When you put these things in binary terms, you get one picture - can vaccinated people still get sick? yes. can they possibly have enough viral loads to infect others? yes.
But when you put them in relative terms you get a completely different picture - can they get sick? yes, but at a rate 10-20x less than unvaccinated people. can they have enough viral load to infect others? yes, but at a lower rate and for a shorter duration.