A lot of things are disruptive. Doesn't make them bad or undesirable. We talk about "disruptive" new technologies here on HN all the time and how the best progress is made by disrupting the status quo. I won't claim that all disruption is good, but who's to say a little disruption in the immigration world wouldn't be?
If an unemployed person loses out to an immigrant (legal or illegal), clearly there's a reason. I'd rather have jobs filled with better people than pander to people who are currently more eligible solely because they had the luck to be born in a certain place at a certain time.
Sure, all else being constant, you want jobs to be filled by the best people available. More likely, the market will see to it that the best people per funds spent are employed, resulting in a race to the bottom. And there are downsides and costs to having millions of unemployed, too. You'll end up paying for them one way or another, through unemployment benefits or prisons.
Are you cool paying healthcare and education costs for the immigrant's four kids? His low hourly wage does not come close to representing his costs given a social welfare state. This is essentially what is happening as criminal employers hire criminal aliens: they externalize massive costs to the larger population.
It sounds like the problem here wouldn't be the immigration, but rather that moving between economic classes is difficult for those with no skills. If we could provide ways for any citizen or immigrant to gain useful skills that are in demand, then we would help mitigate the problems of unemployment as a whole and help reduce the spending burdens of social programs.
The reason he has a depressed wage is because he's undocumented and therefore can be exploited. Reduce the friction involved and more people will be documented, and more on their way to full citizenship. Shady employers won't be able to pay less than min/market wage since the now-documented workers have legal recourse.