The problem is herd immunity.
I agree with you that the problem is "herd immunity", but not in the way you think.
The problem is in people's perception of the risk vs. benefit of vaccines, and the phrase "herd immunity" does a lot to distort that perception. It suggests that perhaps the risk to an individual getting a vaccine is greater than the benefit to that individual, and the primary reason for the pushing the vaccine on people is for the greater good of the population. First, that isn't true: for pretty much all the standard vaccines people get, the risk to the individual by not getting the vaccine is greater than the risk to the individual by getting it. "Herd immunity" is really a bonus, in that getting a vaccine reduces everyone else's risk of getting the disease as well. However, no reasonable parent is going to subject their child to a risk of harm if the sole benefit is to other people's children, and so placing too much emphasis on "herd immunity" really could be doing more harm than good as it could distort the public perception of the benefits of vaccines for the individuals getting them.
A second problem is the terminology itself. As anyone in advertising will tell you, word choice can have a profound psychological effect on people's perceptions. The word "herd" in all other usages of which I'm aware applies to livestock, such as cattle. When doctors talk to parents about "immunizing the herd", it suggests, even just subconsciously, that health care professionals see children as livestock, and not human beings. While this may not be true, if parents are already wary of vaccinating their kids, the phrase "herd immunity" certainly won't push them in the direction of wanting to.