The claim is that the metals are in a stable bond in solution and do not get absorbed by the body. I bet you, though, that when or if they conduct the study they'll find people who opt out are healthier on average, but then say those people are doing other things that benefit like staying away from sugar and exercising. That being said, they might not even admit that because being fit is right-wing nowadays.
Discussion
Thiomersal is very toxic by inhalation, ingestion, and in contact with skin (EC hazard symbol T+), with a danger of cumulative effects. It is also very toxic to aquatic organisms and may cause long-term adverse effects in aquatic environments (EC hazard symbol N).[21] In the body, it is metabolized or degraded to ethylmercury (C2H5Hg+) and thiosalicylate.[15]
https://en.m.wikipedia.org/wiki/Thiomersal
The toxicity of ethylmercury is well studied.[4][1] Like methylmercury, ethylmercury distributes to all body tissues, crossing the blood–brain barrier and the placental barrier, and ethylmercury also moves freely throughout the body.[5]
https://en.m.wikipedia.org/wiki/Ethylmercury
They don't try very hard to make their case, they rely on the general disinterest of the public.
It's either disinterest or willing ignorance