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Have you gotten a chance to read the paper? If not, you should consider reading it. They have explained it. It's got to do around the fact that domestic labor supply is inleastic but the foreign supply is infinitely elastic. You might want to focus on sections VI and VII for more details around their model and assumptions.

I think the interesting question then moves one level down: how well do those elasticity assumptions describe the actual H-2B labor markets in the study?

AEA leans strongly left. I wouldn't put much store in this.

[citation needed]

I’d push back a bit on the idea that they simply missed it. They actually test something very close to this in the appendix: whether losing the lottery changes the competitive environment firms report facing. They don’t find a statistically significant competition effect