Vaccinated individuals can also carry it. It does not stop spread or contraction. These Vaccines are just supposed to reduce the extreme Symptoms

No they are not. They are supposed to allow the immune system to react more quickly after infection (essentially immediately if you have antibodies).

Kill pathogens more quickly, you'll be infectious for less time, and whoever do you infect will have a lower viral load.

That's incorrect. Vaccinated people are contracting Covid at a bit less than 1/5 the rate of the unvaccinated. See for example

https://covid19.sccgov.org/dashboard-case-rates-vaccination-...

I think it is very likely that vaccinated people are contracting COVID-19 less than unvaccinated people, but because most testing is done on symptomatic people, and the vaccine is known to reduce symptoms, the sampling bias is likely overstating the effect.

Vaccination greatly reduces transmission.

Vaccinations reduce your chance of getting infected and reduce the average duration of the infection. The combination of these two factors thus reduce the R value of covid in vaccinated populations when compared to unvaccinated populations with identical network topologies and behaviors.

Stop? No. Reduce? Yes.

"A growing body of evidence indicates that people fully vaccinated with an mRNA vaccine (Pfizer-BioNTech or Moderna) are less likely than unvaccinated persons to acquire SARS-CoV-2 or to transmit it to others."

Source: https://www.cdc.gov/coronavirus/2019-ncov/science/science-br...