I don't have any stats to back this up, but could it be that immigrants are more law abiding because they have to pass the long and difficult process that verifies them? If the system were to change in such a way that more people can immigrate to the US it seems very likely that a lot of the people who would not be granted citizenship would be the type of people who Americans would not want.

If we look at France or Sweden, two countries who excepted a lot of migrants, we can see that the results are not so good. Sweden is now ranked as the country with the most reported rapes [1] and France has areas where the police are afraid to enter [2] .

Think about immigrations like drugs (medical ones). There are many drugs that can save many peoples lives but are not able to pass the FDA for whatever reason (eg. lack of money). While it is sad that people will die because of that, if most drugs were allowed to be marketed by doctors, many more people could die. Similarly with immigration, currently a lot of people who have the potential of adding a lot of value to America are not allowed in. However, if most people were allowed in, it could cause a lot more damage to America.

[1] https://www.gatestoneinstitute.org/pics/large/932.jpg [2] https://en.wikipedia.org/wiki/No-go_area#France