And do you really think that very many people crossing the border actually have good documentation? Do you really think a lot of those people want background checks done on them?

Yes and yes. 100%. Almost all of them will have ID, because if you cross with no ID you're at the lowest priority for being discharged during the pendency of your asylum hearing. The more documentation you have, the more likely it is that you can support your claim of who you are, what country you're from, and that you don't pose any greater risk to the U.S. than the typical Guatemalan.

How hard is it to learn the right key words to say? It's not.

Yes, it is. You're in detention until your credible fear interview, and they take the credible fear interview seriously (in no small part because they're being crushed with people right now). There's zero chance our Saudi national could pass a credible fear interview passing as a Central American asylee; and if you're claiming to be a Yemeni national (or some other nationality not typically seen at the border), you're going to get a much closer look. Especially if you're travelling with no ID.

They don't need to. They can just walk over the border and if caught, claim asylum.

And again, languish in detention until they get their credible fear interview, and either fail that and get deported immediately or pass it and stay detained indefinitely. Because they don't have any documentation of who they are, and speak Arabic instead of Spanish.