When is someone rehabilitated, though? Must someone's past mistakes haunt their life to their grave?

Here in the UK we a sex offenders register, but general crimes get struck off the official register after a certain period of time and no longer come up on background checks. It's not perfect, but it's a lot better than locking people out of work forever!

Are those not exceptions? How many of the unemployed young men have criminal records concerning sex offense or financial fraud?

The criminal mind uses the means available to him. Bernie Madoff was as much a psychopath as a your average mugger; he just had capabilities to not to have to resort to that crude a level of harm.

I would disagree with the implied dichotomy of 'criminal minds' and 'non-criminal minds', especially given the somewhat arbitrary nature of what we as a society decide is a crime worthy of prosecution. Possessing a bit of home-grown marijuana for personal use can give someone a criminal record the same as other more heinous crimes, and many employers will find it easier to blanket-reject anyone with a criminal record rather than dig into the context of each applicant.

Is the answer to not allow any sort of background checks? I would say no, but it also seems like there is a lot of room to improve our current system and find it in our societal heart to not perpetually punish a person for a past minor offense.

You've said why you don't like it but not why it wouldn't work.

It's pretty straightforward to extrapolate from his examples: people do not want to work with nor do business with ex cons. Therefore businesses that hire lots of ex cons likely "wouldn't work" because customers and employees would be scarce.