Visas but not voting for undocumented immigrants

For undocumented immigrants in the U.S., it seems that Americans have four chief concerns: cost to U.S. citizens, safety of U.S. citizens, respect for U.S. laws, and plight of immigrants if they cannot stay in the U.S. I can’t speak to cost. Regarding the other concerns, what if we adopted the following policy:
If an immigrant has crossed the border as an adult or near-adult (i.e., voluntarily) without proper documentation and without a valid asylum claim, that person would never be permitted to vote. The denial of the right to vote would be the penalty for breaking the law and jumping the immigration line. It would also disincentivize a political party from opening the border simply to recruit future voters. However, if that person’s future actions are law-abiding, he or she can stay in the U.S. on a work visa, and in time and with a clean record, that person would be eligible for permanent residency.
It seems like that approach would balance concerns over safety and respect for U.S. laws with concerns over the plight of immigrants. I also think that approach would be acceptable to most Americans.
Thank you for listening.