There was a period of time where we had unexamined DNA evidence - when we knew to collect it but either did not trust it or it was too expensive to do a full analysis of it.
Later on it got cheaper and we understood it enough to trust it more (it is not 100% trustworthy - not how we use it).
They found that 95% of the guilty cases where we had unexamined DNA evidence were in fact guilty. 5% of the time, a person that could not have provided that DNA was convicted.
Which works out to one in 20 convicted people being innocent.
As for the idea that a third found innocent being guilty - we do not find people innocent, we find them "not guilty". And that is pretty rare - prosecutors have a good idea of whether they can win the case and do not waste their time if they cannot. More than 90% of people that go to trial for a serious crime are convicted of something - if not all things.
I find that to be a much better estimate of around 1 in 20 convicts being innocent and a similar amount of those fond not guilty to be guilty. To me, those odds are reasonable effective. I am not outraged at how bad our legal system is, I find it acceptable.
That said:
1) We can always make it better - the bond system for example is vile and evil, keeping innocent poor people in jail without trial, while at the same time letting wealthy criminals get out of jail and commit more crimes.
2) Given the knowledge that about 5% of convicts are innocent, the death penalty is totally misused. If we can't fix it, it should be abolished. I am not against it in theory, but our current use is vile. Prosecutors tend to use it based on the victim and the horror of the crime - without even considering they might have the wrong guy. Kill a black male prostitute, you get jail. Kill a cop or a pretty blond girl, you get the death penalty. They even use it on felony murder cases where your partner killed someone but you did not.
In particular, I would never allow the death penalty to be used when evidence is 'missing',