Here's the thirty second explanation for why subtraction "should be" saturating. First, suppose that we have $a, b, s \in \mathbb{N}$.
The subtraction $a - b$, when it exists, is the solution $s$ to the equation $s + b = a$.
Truncated subtraction is the supremum of the set of solutions to the equation $s + b \leq a$.
That is, when $b > a$, no solutions $s + b = a$ exist, and the supremum of an empty set of natural numbers is 0. Otherwise the supremum will just be the subtraction.
So saturating subtraction is left adjoint to addition (i.e., $a - b \leq s \iff a \leq s + b$), and is hence a natural definition.
Predecessor is just subtracting 1, so it should be saturating too.
I learned about this from Martin Escardo, who also recommended Lawvere's Metric spaces, generalized logic and closed categories as a nice exposition of this and related ideas.
I should add that this adjunction is nice when doing interactive theorem proving. Proofs involving subtraction typically require case distinctions (i.e., you need to case on whether $b > a$ when proving someting about $a - b$), and furthermore subtraction does not have good associativity or commutativity properties.
In contrast, addition on the natural numbers behaves much better, and you can often use this adjunction to massage a proof obligation into a form which is easier to work with.