You concede the fact that the current immigration system in place is broken. Most Americans can’t even pass the citizenship test. These people need our help to literally feed and clothe their families, of whom have almost no opportunity to make a reasonable life for themselves in Mexico. They aren’t looking for a hand out, they want to earn what they get from the US. The biggest problem is not with people getting in, it is with our subsequent treatment of them as something less than a person. It is appalling the way in which people who may be economically lower or less fortunate, simply due to location, are met with a scoff and told to stay in poverty even when we have more than enough to share. If we let them become citizens, we can allow them to be not only productive members of our society but also taxpayers.
America is founded on immigration, as you said, and the opportunity to make a better life for you and your family. If you were born in Mexico or another third world country I bet you’d be singing a different tune.