A lot of this "we don't care about criminals" attitude is cover for many Americans' feelings around race. Despite being only 12% of the American population, there are more black people in prison than whites (though this ratio is improving). Incarcerating black people has a long history going back to the post-Civil War environment when some states created a web of laws designed to incarcerate black people, because they were no longer allowed to own them as slave property.
So when you hear people say "well, Floyd committed a crime previously," it's the long shadow of the past of (1) criminalizing them and then (2) dehumanizing them, in order to treat them as low value chattel - justifying killings or other mistreatment.
So when you hear people say "well, Floyd committed a crime previously," it's the long shadow of the past of (1) criminalizing them and then (2) dehumanizing them, in order to treat them as low value chattel - justifying killings or other mistreatment.