IMO all most of them want is a chance at a better life for themselves and their families.
I believe you are absolutely right. I have known and worked with a few illegals and this describes all of them perfectly.
When I was in college I worked summers for the Forest Service in Wyoming. One of my jobs was inspecting tree planting contractors, and even back then many of the crews were illegals. It was very hard dirty work, not a lot of legals wanted to do it. I spoke a little Spanish, very little, but more than the other Forest Service employees so I got assigned to the Spanish speaking crews. In my experience, on average, they were harder and more reliable workers than the legal crews, so easier to work with. I felt safer amongst them than amongst the white legal crews, the ones who looked more like me. As I got to know them I liked most and even helped them evade immigration in one incident. I now have mixed feelings about what I did, but I do think I have a better understanding of who the illegals are.
The reasons we need to control our migration should not be tied to the "goodness" or "badness" of the people involved. I of course do not condone allowing criminals into the country, but other than entering illegally in my experience most illegal immigrants are good people. I do however believe that we need to control our population growth and who we let in. Some good people may not get in, we can't let all the world's good people into the US.
The asylum seeker thing is a real problem. Too many people can legitimately justify oppression and danger in their home countries. However opening our gates to these people creates a couple of problems, for one it limits our ability to control migration. However at least as importantly it creates a drain on the good people in the countries they come from, perhaps aggravating conditions at home. If we had not let so many good Cubans in I suspect Castro and the communists would not have been so successful in holding on to power. Iran is another example, if we had not let so many Persians in I believe Iran would be a better country today. I have known a lot of Cubans and Persians, all good people, some of the best and brightest I have known. They have been good for the US and good for our economy, however they might have been better for their home countries... Anyway right now the only solution I can offer is that I believe we need to look very hard at what we are doing.
In the end I believe we have to regulate immigration to our benefit, not to the benefit of others. I know that sounds hard, but the world has too many people who want in, can make good cases for getting in, and are good people for us to take. It is just reality and it will get worse so long as our economy and system hold up and world population grows and gets more mobile. It is something we need to deal with rationally, the sooner the better.