On the point about whether British-born claim benefits...
I think the idea is that a given size of population has a given distribution of people in work, in education, on benefits and in retirement.
If immigration increases the numbers of people in work in proportion to the increases that it also causes in the other groups, then as a whole, the system works financially, though there are still potentially problems with capacity, ie running out of housing stock, school places, etc.
If it's out of balance, if more immigrants work than we would expect, the country gets wealthier, and if fewer work, the opposite applies.
I guess the problem with a contributions based system is that, if we refuse to fund a given person who has moved here, they can be sent back to their country of birth. We won't see if they or their children live in poverty over there. But if we withhold funding from teenage mothers, as one poster has suggested, we can't deport them, as this is where they were born, so we would have to get used to seeing that level of poverty in our own streets.