Which one of the sentences below is correct both grammatically and conceptually?
My teacher said that most of it was correct, with only one exception.
My teacher said that all of it was correct, with only one exception.
Grammatically, I would say that both are correct, depending on the situation.
My teacher said that 51-99% of it was correct, with only one exception.
My teacher said that most of it was correct, with only one exception.
versus
My teacher said that 100% of it was correct, with only one exception.
My teacher said that all of it was correct, with only one exception.
As you can see, "most" is less than all, but more than half (which is how it gets to be most) whereas "all" is obviously the sum total.
Conceptually, however, you're right: The second one is weird.
How could it all be correct and yet have something wrong?
But if we consider it from this perspective:
My teacher said that all of it, not including the exception, was correct.
My teacher said that all of it was correct, with only one exception.
Then, yes, we could have it "all" be correct because we had not included the exception within that "all."
What do you think? Does this help?
If you take a look at the example sentence in definition one for the word "exception", it may help make things more clear as well.
The following statement is correct:
My teacher said that all of it was correct, with only one exception.
If you wish to use "most of it", then the sentence is better framed along the lines of:
My teacher said that most of it was correct, the only exception being (go on to elaborate what was wrong).
This is not the same question, but a related/similar one: https://english.stackexchange.com/questions/73649/other-ways-of-saying-except-for-one