That's not correct. If it were true then there would not be high case numbers in countries with very high vaccination rates, but there are.
UK data shows that vaccinated people are more likely to get COVID than unvaccinated people at this point. You cannot argue that vaccination makes you less likely to catch it with figures like that - it's just a made up talking point to try and justify deeply unwise and unethical mandates.
"But you can't be a vector of transmissible diseases" what exactly did you not understand about your own statement? Are you allowed to be a vector or not?