To some degree, you're right. But the problem in your reasoning comes in when the first crime (or strike) was for a lesser crime. For example, if someone commits burglary but later moves up to armed robbery, they have gotten more violent and maybe the escalation in crime and the fact that they have shown an inability to stop committing crime after the first time needs to be factored in. But at the point when all they had committed was the first offense, there was no reason to punish them more heavily because they had not yet shown the escalating violence and inability to conform to the law on a repeat basis.
With armed robbery, the answer may be to just punish all armed robberies, be they a first, second, or third offense, more seriously. But what if the third crime is a lesser offense than the first two--there is still a pattern there and an ongoing disregard for the law.
In a perfect world, I agree with you that the punishments for each individual crime would be enough to stop any future crimes. But unfortunately, even in states that have harsher sentences, repeat crimes happen.
You may be putting limits on the sentence for burglery that I am not. I would make that a minimum of five years. That would be for breaking into someone's house without a gun.
And the jail would be a compound, not a modern prison. The necessities of life would be there - food, shelter, washing facilities. But, it would be a long boring 5 years with no TV, no weight rooms. Clean safe and the most boring place. The cooks would prepare only English food (except no fish and chips).
On the armed burglery, it would be 10 years of the same.
It works in other countries. It can work here also. States that have "harsher" sentences does not mean they have sentences that are harsh enough.
Take Singapore. Very low crime rate. English law and a fair trial. Most people "don't do it" because they know the consequences are harsh. The ones that do, they get tough sentences and don't do it again.
If people are repeating crimes, then sentences and the prison conditions simply were not harsh enough. There is nothing complicated about it.