You're right that overstays are a big problem. Now imagine how much worse it would be if every state got to choose who comes in. The federal government tries real hard to issue visas only to people who won't break the law. They're not entirely successful, but the bar is decently high.
The Syrian refugee hypothetical would have a different problem altogether. The opposition to admitting refugees isn't economic, it's about terrorism. If you can't stop a terrorist posing as a refugee from traveling from California to Texas then the only way to keep Texas safe from that person is to keep him out of the country altogether.
(To be clear, I think the worry over terrorists posing as refugees is just paranoia. But the fear is real.)
Like I said, the federal government would handle screening. States just admit a number of people, they don't vet them- the fed has primacy there because of national security and immigration control. Regardless you can replace Syrian with Mexican and probably should. I only said Syrian refugees for the sake of variety.
But again, as a national security concern it isn't a problem. As long as the number of state green cards is less than the number of visas granted, a terrorist will be much more likely to get a visa and overstay it if necessary.
How exactly are overstays a problem? It's just a person existing in a geographical area longer than the Mafia controlling it would prefer.
All this "theorycrafting" about how best to coerce people for the greater good is silly.
Blah-de-blah-de-blah the government this and the states that and then "we" make everyone do X and Y because Z and then there will be much rejoicing because people have been successfully coerced for the greater good!
Like if I show up at your door each day and force you to skip on one foot for 5 minutes, at gunpoint, that's good because you might not get enough exercise otherwise!
Anyone can see that would be crazy, but when you talk about some huge, gray abstract masses of people, then it's just fine to intervene in their lives in countless different ways.
How about "we" protect people from becoming drug addicts through a certain harmless "gateway drug" by threatening them with life-ruining prison sentences for using/possessing said gateway drug?
Oh wait, "we" tried that already. It didn't "work"[1], and actually WE had no say in any of it!
[1] Unless, of course, the real goal was to hand everyone else's money to the prison industrial complex, in which case it worked splendidly!
The Syrian refugee hypothetical would have a different problem altogether. The opposition to admitting refugees isn't economic, it's about terrorism. If you can't stop a terrorist posing as a refugee from traveling from California to Texas then the only way to keep Texas safe from that person is to keep him out of the country altogether.
(To be clear, I think the worry over terrorists posing as refugees is just paranoia. But the fear is real.)