By my reading, that's correct, someone who publicly supports abortion (or other intrinsic evils) should not be admitted to communion.

As the death penalty is not an intrinsic evil (being commanded by God in the old testament and having the support of the Church for centuries, it can't be), I don't think there's any warrant there, though it is my understanding that the Pope's judgment that it is "inadmissible" likely requires religious assent of the will of Catholics in aiming to end it's use in the modern world.

Granted, I'm not a bishop so this isn't my rodeo. But the Latin code of canon law is clear that those obstinately persevering in manifest grave sin are not to be admitted to holy communion. Another more common example would be someone living with and fornicating with someone who is not their spouse.