I skimmed this and read a few sections closely. Most econometric papers spend too much time rambling on about stuff that most people who actually seek out and read these things already know. Anyway, +2.6% in wages for non-college educated natives, over 20 years, is not a ringing endorsement. I honestly doubt anyone would notice this, and over two decades that's just noise, frankly. If I'm understanding one of the central claims correctly, it's not that low skilled immigrants "took" jobs from natives, it's that natives ended up working more, and as a result their earning power increased. That's ... also not a ringing endorsement. Sure, you didn't get fired in favor of an immigrant, but you had to take a second shift to stay in the game. Wow what a benefit.
I'm also going to quibble a bit with how they constructed this Instrumental Variable. The way it's constructed, the higher the turnover (for lack of a better term) from native to foreign, the better the predictor it is (because in their theory, it's exogenous to wages at the local level). Does anyone really believe this? Immigrants respond to incentives just like anyone else. They're not choosing to immigrate to Sac City, Iowa, or any other low-COL/low-wage town, for a reason. It also doesn't answer my original question, which was central to my initial claim: why? The "why" is going to answer a lot of other layered questions about hostility to ever-increasing immigration. Are firms moving in that exploit low-wage immigrants, generating other adjacent economic activity? Probably! Not controlled for or referenced at all in this, and this paper is by no means a definitive conclusion that high low-skill immigration is good (even on a two decade timeline, which even typing that is just absurd).