Sell access to the United States to the highest bidders? Why not? Given the extent to which filing fees have gone up, we're pretty much doing it already. And, since the immigration system is self-supporting through filing fees, maybe that would create a budget for enforcement big enough that people would stop fretting over whether we're "tough enough" on "illegals," assuming that there would be any "illegals" under a system like this. (Probably, there would be. There will always be a need to keep out those who would put individuals or whole populations in harm's way, and those people will always try to find a way in to do harm.)
I'm not sure I'd make the system as simple as writing a check. There should still be some evidence of actual or potential employment or, more generally, some set of skills and/or assets that would ensure that the person was not automatically going to become a public charge. Then, too, there's the question of how big a check to require: I think this should be means-tested, based on the person's resources and plans for using them.
In any event, as FDR said, "try something," and this might be worth a try. It can't be worse than the status quo.