All of these social programs suddenly become much more popular when they are restricted to the majority population. There have been surveys demonstrating this. Another way to see it is to point to places with a lot of those policies, one of the first rebuttals you will receive is a yarn about the homogeneous population, or at least one far more so, in those places.

So, it isn't that they think they won't be covered, it is because so many think it will cover people they don't want it to.