Why would them leaving be viewed as a "broken" immigration system? I'd say it's broken if we don't enforce immigration laws, not if we enforce them.
It would be trivial to set up a system to validate the right to work in the US. In fact one already exists - E-Verify. But many groups have resisted the national roll out and penalties if businesses don't comply (similar to how we have sanctuary cities).
The funny part is that in all the other countries I've lived in, this issue never comes up. Sure, there are illegal immigrants, etc, but nobody questions when they are sent home or when businesses are fined for violating immigration law.
I'm not sure why it's such a contentious issue in the US.