Why would a country not deport non-criminal illegal aliens? Not being American and having lived in 4 different countries, I’ve never encountered this attitude except in the US.

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.