I am not sure I follow your statement of how receiving this driver's license makes this no longer an issue of their immigration status. They are still very much illegal immigrants, and receiving this ID does not provide them any "legal" status in the US. That is why such issues are so confounding - it is granting legal privileges to illegal immigrants, which is ironic to say the least.
The point you made that I was responding to was that there was an increase in hit and runs because illegal immigrants who were involved in car accidents couldn't stop to show ID or insurance and thus drove off.
If you give the illegal immigrants a way to get ID and insurance that doesn't carry with it the threat of deportation in and of itself then this reason for them driving off disappears. They have ID and insurance, therefore they can show it. Thus, if they continued to drive off after an accident it's not longer a matter of their immigration status... after all they can have ID and insurance even with their status... it's a matter of, in essence, morality and criminality.
I think (or at least, would hope) that all Americans would agree that in an ideal world, all immigration would take place through legal channels. The question Americans need to ask is whether or not we are "early enough" in the game to be able to tackle the issue of illegal immigration head on through targeted deportation, or pass laws to adapt to an existence where illegal immigration is a fact of American life.
From what I understand, estimates in 2010 put the number of illegal immigrants at over 10,000,000 in the US... and I imagine it's increased by then, even with the Financial Crisis causing many to return.
The cost, time and effort to find, let alone deport, even a significant number of those is going to be astronomical and frankly isn't realistic to even consider. The balance to be struck is between finding the best way to accommodate those already there while also preventing more from arriving; be it by tightening security or making legal immigration an easier and better option.
There are legal avenues for applying for refugee status and asylum. The US has refugee quotas and a designated application process. Illegal immigrants bypass this entirely, which is essentially the crux of the issue. There are programs for unaccompanied refugee minors who essentially land unattended on US soil, but I think we can all agree that this privilege is being abused to no end at this point (given that many parents are intentionally sending their kids on these treks across Mexico for a "better life" in the US). What message are we sending by turning this token of good will into a regimented immigration avenue?
That's sort of my point though... there are very different systems in place for dealing with refugees and asylum seekers compared to illegal immigrants. Using the example of refugees (either as a reason of why illegal immigrants should be treated less harshly or to use as an example of someone who deserves assistance compared to say economic migrants)in a debate such as this is somewhat awkward because they're two separate issues, albeit two that are often conflated.