I'm not clear on why you don't regard ZFC as an example. You say:
Weakening the axiom of naive comprehension has not been a popular way of escaping from the set-theoretic paradoxes because no consistent weakenings seem to be particularly well motivated or even to lead to understandable models.
But it seems to me that the ZFC axioms of set theory result essentially from a weakening of naive comprehension, are highly popular, are well motivated and seem to avoid the paradoxes while having an abundance of understandable models.
In particular, the ZFC axiom of separation is the result of weakening the naive comprehension axiom to the assertion that for any property $\phi$ and any set $A$, the collection $\{ \ x\ \mid\ x\in A\text{ and }\phi(x)\ \}$ forms a set. And one can similarly view the replacement axiom as an instance or weakening of naive comprehension, asserting of any set $A$ and property $\phi$ that the collection $\{\ x\ \mid\ \exists a\in A\ x\text{ is unique such that }\phi(x,a)\ \}$ forms a set.
Furthermore, the ZFC formulation of set theory seems to be very well motivated by the iterative conception of set, where one views the class of all sets being formed in a well-founded cumulative hierarchy formed in stages, in which the elements of a set are constructed at earlier stages than the set itself, and the stages continue in an endless transfinite progression. In essence, one must construct the elements of a set before constructing the set itself. On this philosophical view of how sets are formed, there is ample support for the separation and replacement axioms, and essentially none for the naive comprehension axiom (since there seems in general no reason to suppose all the $x$ with the desired property exist by some stage).