It's really hard to come up with a system to fire judges for decisions you don't like that won't be abused against judges you like who do things that are controversial, but in your eyes necessary.
One underappreciated point is that Trump's insistence on a special master shows how atrocious his actions have been. He has allowed classified documents, which are supposed to have a well-documented chain of custody, to be intermingled with his own personal documents.
It's a bit misleading to suggest that the conservative response is that "they were probably illegal."
As a factual matter, the majority are almost certainly illegal (or to put it more politely, have not been granted legal citizenship and do not meet the strict standards for asylum.)
I do suspect that the existence of sanctuary cities, where the policy is to not enforce immigration laws, serves to attract undocumented immigrants, as an example of how inconsistent enforcement of existing immigration laws is.
The existence of sanctuary cities isn't the only thing that makes border enforcement more difficult. I also mentioned actions by the federal government.
A major one would be the Biden administration's decision to end the "remain in Mexico" policy.
https://www.pbs.org/newshour/politic...-mexico-policy
The DHS pushed a new policy for ICE to end raids on workplaces where there were undocumented immigrants.
https://www.npr.org/2021/10/12/10452...lace-raids-dhs
ICE officials were ordered not to prosecute anyone who had been in the country for twenty months, which led to 60,000 case closures.
https://www.washingtonexaminer.com/p...-facto-amnesty
The question about whether federal guidance to deportation officials violated federal laws will be addressed by the Supreme Court.
https://www.texastribune.org/2022/07...supreme-court/
You may think these are wise policies but that's different from whether it makes it easier or harder for states to enforce immigration laws.