I think what irritates the hell out of pro-vaxxers is that anti-vaxers tend to think for themselves regardless of what -- say -- either the fanatically leftist loyal media outlets or, indeed, Fox News or Trump says about these vaccines.
Since leftists -- and most pro-vaxxers are leftists -- will not and perhaps cannot think for themselves the very concept of NOT following either the orders or suggestions from one's own leadership . . . shocks the holy hell out of them.
But Lindsey Graham is just another guy with his own opinion regarding the efficacy and SAFETY of these various NOT APPROVED by the FDA innoculations to combat a virus with a 99.8% recovery rate. The fact that he is also nominally a conservative is a non-issue among people who can and will . . . think for themselves.