You have it exactly backwards: had the GOP passed that bill it would have codified thousands of “asylees” at the border per day…incentivizing tens of thousands more to head north in the hopes that they had won the lottery for that particular day.

It would not have codified any number of asylees. It would have only set a threshold at which time the President could exercise a power that he did not currently have - to apply expedited removal procedures to asylum claimants that under current law are not subject to expedited removal. That would have created no additional incentives. Currently there are no provisions for expedited removal for that category of border crosser.

The only change to the status quo was to reduce the protections for asylees. It put a high daily number on when that reduced protection would kick in, to be sure - but the only actual effect was negative for asylees.

Conservative media falsely portrayed that as codifying a benefit for asylees, when it was the exact opposite, in order to drum up opposition to the bill. But had the bill passed, Trump would have had vastly more resources to use to actually increase deportation rates. The high threshold for the expedited removal process to kick in wouldn't have affected him in the least - it only gives the President more removal power, just in rare situations. But the GOP forewent the extra resources because it was far more politically useful for them to do so.

So now they're paying for that choice. The Trump Administration doesn't have the tools to accomplish one of their highest priority goals. They can't move any more people through the deportation pipeline, because they don't have the agents and LEO's and lawyers and immigration court space and detention facilities to get it done. They're trying to divert resources from other agencies (like DoD and other law enforcement agencies), but that's not really working. The people aren't trained to do this kind of work, so bringing law enforcement personnel from other parts of the government doesn't really add much.

It's exactly what we were all saying would happen. The rate of deportations (which conservatives found unacceptably low) was a product of resource constraints, and it couldn't be materially increased just by having more of a hardliner setting policy. So now you have the most hardline administration you can imagine, whose immigration policy is led by Miller and Homan who are whipping ICE as hard as they can, and the deportation rate hasn't budged. Again, because the GOP turned down the resources back when the Democrats were offering them.