Maybe what they meant by "emotional" is that a lot of antivax sites tell stories of kids who were allegedly injured by vaccines. These stories, even if true, are anecdotal and not statistically relevant, but are of course emotionally relevant and emotion-charged. Whereas with pro-vax sites, they don't really do that as much. At least they didn't used to. I am seeing more personal stories of kids who died or were injured from VPDs, but that's only recently. And certainly not on government sites, like the CDC or AAP. Whether that's fighting fire with fire, or using a relevant real-life example depends on how you look at it, I guess.
I think that a lot of the reason that anti-vax sentiments are thought to be emotionally-based is because at first they didn't HAVE a lot of science, they had to rely on emotional, real life examples. I dont' meant that in a deregoatory way at all. In fact, IMO, it was these people who spurred on a lot of the studies into the safety of vaccines. They didn't have the science because it hadn't been done thoroughly yet, and so their request was for that to change. Whereas provax people at that time were using arguments comprised of charts and epidemiology. Relevant, but not terribly emotional.
While I don't think that the emotional appeal should override the scientific evidence, I think it's a useful way of conveying the impact of their decisions to people. Whether that's from showing a child who has been allegedly severely damaged by vaccines, or showing how a baby who has pertussis struggles to breathe. I think for most people, it just makes these things more real, yk?