We made these decisions for various reasons, but broadly because the voters felt that US citizenship and lawfullness should be presumptions, rather than something you had to prove in order to enjoy your rights as a citizen.
For an immigration agent, this is really tough. You have to identify unauthorized immigrants in an environment where you can't just require lawful citizens to carry ID or proof of citizenship. You legally can't arrest or (more than briefly) detain a US citizen for failure to carry citizenship documents. You have to walk on eggshells even with actual unauthorized immigrants, to avoid violating the law. And our proof-of-identity document systems are deliberately decentralized and unreliable, so you can't just check a master database. It's a tough problem!
But that's the way the cookie crumbles. We designed our society to make this kind of "papers please" enforcement difficult, which means that immigration enforcement needs to be smarter and more savvy, or else we need to actually change the laws. What ICE and CBP are trying to do now is just to ignore the law, and that doesn't work. Citizens' built this law to protect their rights; you can't just take away those rights because CBP have a tough job.
My only point is that when a deportation order shows a name and face, people can still produce fraudulent documents showing they are a citizen.
It’s not a uniquely American problem.
If you want a national estimate that grossly undercounts, just multiply by the 20 field offices. Now we’re in the hundreds of thousands.
If the paperwork is that easy to duplicate, it's on the government to make it more difficult, not beating in the faces of citizens until they produce documentation they don't have.
Hundreds of thousands of aliens with fraudulent documents seems like a huge problem to me.
Keep in mind it’s US law for every alien to keep documentation of legal status on their person at all times (US code 8 USC § 1304(e)).
Since we know fraudulent documents are not uncommon, then immigration officials must have some powers to validate a person’s status if uncertainty remains.
Obviously in case (1) you can just detain and carefully identify people, since they're actually doing dangerous things and you'll have probable cause to detain them. Note that this was the case that nearly all the 2024 election rhetoric was focused on (i.e., deporting the "rapists and murders".)
In practice we're seeing that the true goal is (2), rapid emergency deportation and arrest of non-criminal immigrants. There is absolutely no emergency that requires this to be done quickly or carelessly, such that there's any risk to US citizens' (or immigrants) civil rights. It can be done carefully, with strong evidence, without violating civil rights. But of course, violating civil rights and creating disorder appears to be the goal.
Controlling who enters and stays in your country is just table stakes for a functioning state. Without it, you don’t even have control over basic things like security.
In every other country I’ve lived in anyone who enters illegally or overstays is promptly removed.
So to answer your question, fraudulent documents is a massive problem. Now not only do you have no control over who enters and stays you have no ability to even determine who these people are even if you do encounter them.