I guess part of the problem is that even if the disease is mild to the vaccinated people who do catch it, they could then pass it on to someone else, who for medical reasons may not be able to be vaccinated, and lead to severe disease in them. That’s why the emphasis on being able to prove a reduction in transmissibility to reduce the chance of that happening as much as possible.
My uneducated guess says if the vaccines have been proven to cause a reduction in severity of the disease, it would seem to follow that there is a corresponding reduction in the shedding of viral particles = reduced transmissibility, but that’s purely my non-scientific reading of it.