That's right. The proponents of minimum sentences say that they deter crime, that the main good they do is they show how serious we're taking the offence, and it will stop people. Of course, with punishment after the fact, the harm has already been done.
The main thing about minimum sentences is, yes, it increases prison populations. Yes, it's expensive and it takes time in court, and it will stop people from committing crimes, but that's not how deterrence works. That's not how people are deterred. If you are committing crimes because you don't care or because you have a mental health issue or an addiction issue, deterrence does not work. Just jacking up the penalty doesn't stop people from committing crimes. A sentence like that, in addition to unfairness, can result in cruel punishment in some of those more unusual cases.
For some cases that may fall outside the norm and have compelling facts behind them, the increase in sentence can cut people off from communities and can crush an individual. That is part of the reason we see rates of recidivism. The rate of people committing crimes again goes up when we see sentences imposed under the minimum sentence regime.