"We have a broken immigration system. Nobody denies it."
That's misleading because the problem isn't with our rules and laws so much as the fact that they aren't enforced. There wouldn't be a problem with illegal immigration if politicians would simply allow our laws to be enforced. If you consider "system" to including not just our laws but the enforcement of those laws, then the statement isn't misleading. However, since it's almost always said in the context of supporting changes to those laws, it is misleading.
This talking point is like a shady mechanic removing all the spark plugs from your car's engine and then saying, "see, your engine doesn't work."