But there is doing what's right and not what's wrong. If somebody is in a country illegally then there are legally only two choices.
Either that person applies for some form of legal status or they don't and they're deported. The law doesn't allow for some utopia where people exist somewhere illegally.
It doesn't benefit migrants either because they're technically ineligible for any state benefits, healthcare, voting or anything whatsoever. The only single benefit is that it benefits what you're describing which is actually an illegal economy.
it's actually easy to resolve, give/impose on the illegals (pick whichever suits your viewpoint) a temporary visa allowing them to work and live in the USA for a period of, let's say 2 years, during which time they must apply for full residency. This gets them in the system, gets them paying taxes, gets them access to benefits and healthcare and gives them a route to a "better life". USA retains the benefit of the workers.
Failing to register becomes a deportable offense within oh I don't know, 6 months, meaning you should then have a smaller issue to deal with.
Of course to make it attractive, the permanent residency should be readily achievable, rule out those with criminal records for violent, sexual or class A drug offenses, let the rest in.
Obviously you'll have to absorb any economic impact of higher wages, higher social security costs etc, but hey, there's no such thing as a free lunch is there?
This politics lark is easy , Vote Flook for PM/Prez