Meanwhile, among the many people I know who have been vaccinated, there's been just a single breakthrough case, and that case was due to catching covid from an unnvaccinated co-worker.

Since you seem a bit confused, here's what we know and don't know AFAIK:

1) Vaccination reduces your chance of getting infected. The efficacy depends on the strain of covid and the vaccine, but I have yet to see any science indicating that any vaccine has no efficacy against any strain.

2) Vaccination may or may not reduce your infectiousness while you are sick

3) Vaccination does reduce the average duration of infection, which means that while you may be as infectious as a non-vaccinated person while you are sick, you have a shorter window in which to spread that sickness.

So despite the unclearness of #2, we are very clearly able to say that vaccines reduce transmission because of #1 and #3.