The vaccines were not approved on the basis that they reduced transmission. This was a widely held hope, given that other vaccines do this, but that wasn't an endpoint of the studies. There was a little bit of data in the Moderna trial that was very encouraging, namely that everyone in the trial took a PCR test at the time of their second dose, and people in the active arm had a much lower positivity rate. There was also a lot of epidemiological evidence for the vaccines dramatically reducing transmission.
The problem is that this data was gathered with a different version of the virus than we have now, and it was gathered with relatively fresh vaccinations. Even pre-Delta, you could see immunity waning leading to increased transmission in Israel. With Omicron, the efficacy of the vaccines is also much less.
That said, from all the evidence we have, the existing vaccines still do reduce transmission, but not enough to prevent epidemic spread. There are a lot of numbers, because every study has a different population and different ways of measuring, but roughly speaking, being vaccinated cuts your risk of contracting and transmitting COVID by something on the order of 50%. That's a lot better than nothing, but COVID is so contagious that it's not enough.
The "R values theory" is perfectly reasonable and almost tautologically correct, and works regardless of an animal reservoir and imported cases from travel. We have the same issue with imported cases with Measles, but hardly anyone gets Measles, because almost everyone is vaccinated, and the R value in the population is much less than 1. If you import a case, that case infects less than 1 person on average, who infects less than 1 person on average, etc, until it dies out. RIt would be great if COVID and COVID vaccines were like Measles, but unfortunately the vaccine isn't yet good enough.