As typical of many of these writers, he's clearly showing his bias when looking at the world (namely, from a secular Western sense). The fact that it is failing in the current secular societies does not mean that it is ineffective in other societies, or that it is in itself bad. We don't have the problems he listed in Islam for example.

> but that isn’t the case in capital punishment as someone in custody is no longer a mortal threat to society.

It's not just about being a threat, it's about justice.

And what do you do if a person is falsely accused?

Under Islamic rules, suspicion would fend off the Hadd punishment: https://sunnah.com/tirmidhi/17/2.