A jail, however, is not supposed to serve the purpose of being a mechanism of forgiveness. It is a mechanism of punishment. We as a citizenry give the government the sole right to jail / punish citizens as a way of enforcing the law. While someone may be reformed as an outcome of serving time, it is not the primary goal. Otherwise, we would see a higher focus on lowering recidivism rates.
It is not expressly a mechanism of punishment. It is expressly a mechanism of debt repayment; it can be argued that that is closer to a mechanism of forgiveness than one of punishment.
This depends on the jurisdiction. Prisons serve many apparent purposes, the mix of claimed and actual intents varies: punishment, deterrent, removal from society for society's protection, commercial reasons, reformation, historical precedent ...
Whatever, I agree that few jurisdictions do a great job of lowering recidivism, putting into place the societal structures to support ex convicts after release in finding stable work and so on.
A jail, however, is not supposed to serve the purpose of being a mechanism of forgiveness. It is a mechanism of punishment. We as a citizenry give the government the sole right to jail / punish citizens as a way of enforcing the law. While someone may be reformed as an outcome of serving time, it is not the primary goal. Otherwise, we would see a higher focus on lowering recidivism rates.