D.P. has been over this fallacy to death. The point is that immigrants don't just come and "eat" - to use your metaphor they actually come and start working in the kitchen (they contribute more to society than they take in public services).
Agreed, I employee a Hungarian and a Slovakian myself on fully skilled wages, but the fact that public services are overwhelmed is not something linked to if they work or not, its linked to not enough capacity for the current population. Meanwhile it continues at unchecked levels.
This is my issue with it. To go back to my analogy, if they come on-board to eat or not, they are still on a boat that isn't built to hold them with no idea how many more will be boarding.