For the same reason we shouldn't generalize that all illegals are criminals (beyond the fact that they are illegal in the first place), we can't really base sweeping policy on anecdotal evidence from one or two families that are trying to do the right thing. You said it yourself, how likely is it that legal citizens would do the honest thing and pay their taxes if they could get away without doing it? That makes it even less likely that even a small fraction of the non-criminal illegals are jumping through the requisite hoops to pay income taxes, especially considering they didn't (or tried but couldn't) jump through the hoops to become citizens.
It's obvious that the matter is very complicated and as such will require a complicated (and likely very expensive) solution. It's easy enough (hah, if only) to identify the criminals and deport them. But the vast majority that are not criminals need to be handled delicately. Those like the individual you mentioned shouldn't be lumped in with those that haven't made any effort to do the right thing, in a perfect world anyway. I can't even begin to imagine how to separate them into distinct categories with distinct rights/penalties/paths to citizenship/etc.
But it has to be done. Kicking them all out or letting them all stay (without penalty and with no path to citizenship) are both non-options, the latter of which we've been doing for far too long. I won't reiterate the countless reasons why, but it should be obvious that it is bad for both the legal and illegal citizens in the long run even though it may have been beneficial for each in the short term.
And I think most of us can agree that we're not angry at the immigrants themselves. We're angry at our government for letting it come to this. This country was founded by and for immigrants, and it should be one of the most accepting of immigrants in the world. Should be, but it's not. Somehow our government has made it both too hard to immigrate here legally and too easy to immigrate here illegally. The former creates the perception that America as a country is anti-immigration, and the latter (through the vitriol it causes) creates the perception that Americans as a people are anti-immigration. That is a real shame and needs to change.