You can achieve this insanely easily, too. Just register every immigrant that comes across and give them a temporary 2 year visa (conditional on the normal things like criminal record, documented connections to terrorist groups, war criminal status, and so forth). Over those two years, compare their tax returns to their use of services. After 2 years, either grant a permanent visa or let their visa expire and deport them. If they're either using services or paying taxes we'll have a way to find them, and if they're not doing either, they're obviously not causing any obvious problems.

Once you've made it past the immigration counter enforcement is virtually impossible. Most cities and counties don't want to be involved in immigration enforcement because its disruptive to the community. And it's left to the visa holder to go willingly. A big problem is people arrive on tourist visas and don't leave by the appointed date. ICE can't really do anything about it unless you try to leave through a border checkpoint. This causes an incentive to stay rather than leave.

Maybe without all the absurd, overcomplicated immigration laws to enforce, ICE will have the resources to implement the followup checks and send local agents to apprehend and deport people.