Your native-speaker correspondent is correct. You could write it off as an idiomatic oddity of English (there are lots, after all). Or you could use the following as a conceptual breakdown of "why it works".
The simple present ("gets", "nuzzles") is used for habitual aspect. So when the chatty ex-carpenter's apprentice in the story was talking to the narrator, he would have said...
I get itchy cheeks when a horse nuzzles my face.
(For simplicity, hereafter the narrator will be called "Alice", and the chatty person described will be "Bob".)
Now, if Alice directly quoted what Bob just said, she would say...
Bob said, "I get itchy cheeks when a horse nuzzles my face."
But this isn't what Alice does, and indeed, in speech (and in writing meant to mimic a natural tone of speech), direct quotation is pretty rare. Instead, it's usual to change the person—but not the tense:
Bob said he gets itchy cheeks when a horse nuzzles his face.
Changing "gets" to "got" would lose the habitual aspect. You would also have to change "nuzzles" to "nuzzled", because having the two verbs in different tense is wrong. (Well, okay, by "wrong" I just mean that it reads really awkwardly to me, for what that's worth.)
On that topic, you may have noticed yourself that "nuzzles" is actually Bob using the same principle. It would be possibly to use a longer phrase to indicate habitual aspect without the simple present tense, but it be rather long-winded—something like...
I have always gotten itchy cheeks whenever a horse has nuzzled my face.
Finally, to your "a day before his death" example. It reads just fine to me. But I can also see how a reader (even a native Anglophone!) might be struck by the apparent oddness of the present tense, and find it strange or even funny. This one could legitimately be rephrased like so...
A day before his death he said that he had never tipped waiters.
If it didn't mention his death, though, this could be interpreted as "he has never done so, but he's considering doing so now"!