Once a ticket is paid, what act of illegality is ongoing?
This "not-amnesty" program would be equivalent to telling people who have received parking tickets and not paid them that if they pay the tickets they will be allowed to park in the no-parking zones with impunity.
Of course not, you could let them die or remain sick or etc. Don't blame someone else if you or really we the American public are not willing to let illegals die rather than treat them in our emergency rooms. THAT IS OUR CHOICE.
The choice of the vast majority of Americans is to not have them get sick in this country by not having them enter this country in the first place.