No horse either but here is an attempt (ignoring criminal record as you say): Opening the border and letting her rip is clearly not sustainable in the medium term. So you try to make it (reasonably) hard to get in incl. turning people away at the border.

Once they are in (incl illegally so) you concede you have lost on this instance. Now you admit that forcefully removing immigrants carries too high a cost (literally + damage in the communities you remove the immigrants from + your humanitarian image). So you don't.

Somehow that balance seems really hard to get right and edge cases (criminal record) matter.