Why doesn't Paul sign all of his letters like he says? 2 Thessalonians 3:17 (NLT)
Here is my greeting in my own handwriting—Paul. I do this in all my letters to prove they are from me.

But such a greeting only appears in five of Paul's letters: 1 Corinthians, Galatians, Colossians, 2 Thessalonians, and Philemon. Why, then, do we trust the other epistles without Paul's proof?
The answer might be as simple as this:

It is Paul's final salutation, not the caveat we find in 2 Thessalonians 2:17, which Paul wrote in all his letters. If he did not add the 2 Thessalonians caveat, then we can assume he wrote the entire letter, including the final salutation.
In other words, the final salutation in all his letters was in his own hand, whether he or whether an amanuensis wrote the letter.