Vance did later admit that this was just a rumor, and then encouraged people the keep on spreading it.
BTW, the immigrants in Springfield are there legally, which is another thing Trump has repeatedly lied about.
The big thing though is that they people I'm talking about who base their support of Trump on things like the eating pets things live in areas where there are not many immigrants (legal or illegal). They personally are not seeing any problems immigrant related.
All their information on this comes from Trump. And most of their information on every other issue. And even if they finally are convinced that he lied on something they never question anything else and go try to verify it.
And that's why I would consider them to be not quite up to par in the thinking department.
The example you gave is an interesting one, because it's a situation where a very real concern--the public resources of small communities being stretched beyond their limits because the federal government refuses to enforce immigration law--was deflected by the media to something ridiculous. ("Brown people are invading our towns and eating our cats!") People voted for Trump because they saw through that kind of tactic, and they're far from stupid for doing so.