Vaccination reduced transmission of the previous Kent Variant by 40-50%, I've looked for similar data around Delta, but there's not much in evidence (I work for a rather large Pharma company, though not one of the vaccine providers, and so have access to research data)
There was a large study in France that concluded -
-Vaccinated people with symptomatic covid shed similar levels of virus during the symptomatic phase.
-The symptomatic phase in the vaccinated is however far shorter.
-Vaccination reduces that chances of catching Covid even with Delta so you have fewer Covid carriers around
So even on this simplistic level, it makes sense.
I am however aware of the irony that only 7 EPL* clubs have higher than 50% of their squads vaccinated, these entitled young men have been allowed to operate all through this, yet we've to show vaccination status to watch them.
* Haven't seen such data for our game, I don't imagine it's any better.