My understanding, is that's what the UK does, with an exemption for certain jobs, like teachers and creche hosts. In the US, I think some states have the ability to expunge convictions. Not sure about federal crimes, though.
The "scarlet letter" of a past conviction is a very real issue, and keeps some folks down. People can get past it, though. I know folks that served time for murder, that have very good careers, and people that have misdemeanor records, that have always struggled.