You could once-and-for-all prove the error correcting program correct and then use it again and again for every program. But that wouldn't actually help getting programs correct. For example if your error correcting program was to write every program three times, then if one of the three programs gives a different answer you use the answer of the other two. Now a human writing a program would probably just write the same bug in each of the three versions, i.e. the three versions are not independent. You could have the three versions written by three different humans. But that doesn't solve the problem either because some errors are just likely to be made by all three of them (for example forgetting to check some edge condition somewhere). So again the three versions are correlated. Or worse the problem could be in the specification all three of them got. I don't think there is any way out of this.
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You could once-and-for-all prove the error correcting program correct and then use it again and again for every program. But that wouldn't actually help getting programs correct. For example if your error correcting program was to write every program three times, then if one of the three programs gives a different answer you use the answer of the other two. Now a human writing a program would probably just write the same bug in each of the three versions, i.e. the three versions are not independent. You could have the three versions written by three different humans. But that doesn't solve the problem either because some errors are just likely to be made by all three of them (for example forgetting to check some edge condition somewhere). So again the three versions are correlated. Or worse the problem could be in the specification all three of them got. I don't think there is any way out of this.