The current system of incarceration is actually a part of the problem, not part of the solution. It puts first-time offenders in an environment where they are surrounded by life-long offenders. Part of human psychology is that we tend to normalize the attitudes of those we are around the most, so these first-time offenders start to see the attitudes of the folks they are incarcerated with as "normal". Then society makes it difficult to get a job and get back to being productive afterward, which reinforces some of the persecution mentality that fosters recidivism. The current system tries to do what you said: "convincing them" through punishment ("hard time" is meant to be a deterrent).
Note that I'm not saying we shouldn't have incarceration. I'm just saying it's proven to be a pretty weak deterrent. It needs a change. Small-scale social experiments have shown remarkable results, even with violent offenders. Giving them jobs of trust, teaching them their place in a healthy social order, teaching them skills - these are the things that tend to reduce recidivism. Yet, when programs are suggested for the prison system, people don't want to pay for good education within the prisons, nor for improved conditions, because "they don't deserve that - they should suffer for what they did." It sounds right to most of us, but it is counter-productive for society. It just leads us to spend more money for future incarcerations, not to mention the other indirect costs (higher cost of insurance, more police needed, etc.).
It's a problem that doesn't have an easy answer, but we still need to start making the change. The US is the clear leader in this problem (not in the solution, in the problem, itself), so probably the place where change will bring the most benefit.
Bolded - I would say most have already normalized their attitudes as to their behavioral norms before they were caught and put in prisons. Society does indeed tend to make it difficult for those released from prisons to get good jobs and move up in society. Who do you think has the most blame, society or the offenders, especially considering the recidivism, and which was first, the chicken or the egg?
Underlined - Yes, it seems to have been. But this is not a new problem. I am not aware of the small-scale experiments you have mentioned. I would hope they work. How long have they shown it takes to turn a felon's attitude around to wanting to be a productive member of society? Are they also taught that they don't get an automatic pass to trusted status in society, that they have to work for it, often for a long time, as have those who haven't offended and gone to prison?
Bolded and underlined - I believe many prisons do have opportunity for education and learning skills. Some may not due to overcrowding. But what do prisoners do with that education and those skills when they are released? They cannot expect a free ride into the upper classes of society. Even if you can show studies supporting education and skill teaching, what was done to society to make them happily accepting? That has to be part of the solution don't you think? How was it done?
In all this, I don't mean to sound totally pessimistic and unsupportive of people being able to overcome previous bad decisions, and return to society as happy and productive citizens. In one way, the ideal would be to stop sentencing people to set terms of imprisonment, but rather until rehabilitation. But that idea rather scares me. Who gets to decide when someone has been rehabilitated? Are errors in the decision allowed for? Again, we aren't there yet, and is it even a defensible solution?
I do actually agree that those sent to prisons who do want to change their life and stop living a life of crime, have an uphill battle considering those they are forced to live amongst, and the lack of guard control in many prisons.
That to me would be the first step in giving that type of person a chance. Inmate control of events inside a prison should be prevented. Attempts by prisoners to take any small amount of control should be punished.
I don't know all the good solutions to prison reform, but that seems to be the best place to start. Prisoners have to know that there are societal norms, controlled by society at large, not a prison society. They must be required to follow those norms, including not being allowed to exploit guards, whether by willingness of the guards, or coercion of the guards.
Won't cure all the problems, but I think it would be a good start.