Yes, the welfare aspect is a large concern. A welfare state and open borders do not mix well. However, it feels like you're (dis)missing the post's penultimate point of restricting federal welfare from applying to non-citizens. If non-citizens were ineligible for any FEDERAL welfare, would that help assuage your concerns any? As I see it, this is not only more in line with a decentralized, federalist, Constitutional, 10th amendment perspective of governance, but also blocks the greatest source of moral hazard (federal welfare programs) while still leaving states and localities able to determine what type of welfare (if any) they wish to provide for non-citizens.
Plus, by lowering the barriers to enter legally, is it not likely that we would see far more legal, taxpaying immigrants?
And yes, borders demarcate different govt jurisdictional boundaries. But again, crossing one, in and of itself, is not directly harming anyone.