They do reduce the chance of transmission. Vaccinated people have a lower viral load when infected.
And even if there was no difference it would still make sense to me. There's likely a large overlap between anti-vaxxers and people who don't like adhering to safety measures.
CDC Director Rochelle Walensky said that Covid-19 vaccines are no longer effective at preventing transmission of the virus.
"...what they can’t do anymore is prevent transmission. So if you're going home to somebody who has not been vaccinated, somebody who can't get vaccinated... I would suggest you wear a mask in a public indoor setting,"
And even if there was no difference it would still make sense to me. There's likely a large overlap between anti-vaxxers and people who don't like adhering to safety measures.