So basically, Ballentine does not believe in the quantum Zeno paradox because he does not believe in collapse.
But it seems that he does not understand that effective collapse can almost be "explained" by modern understanding of decoherence, and it seems to be because he is not aware of the importance of decoherence.
The reason for such a suspicion comes from another part of his (otherwise great) book:
Sec. 9.3 - The Interpretation of a state vector
Subsection - The measurement theorem for general states
After Eq. (9.13) he writes:
"The terms with alpha_r1 notequal alpha _r2 indicate a coherent superposition of macroscopically distinct indicator vectors ... It is clear that the nondiagonal terms in (9.13) cannot vanish ..."
But it seems to me that someone who were familiar with decoherence would immediately recognize that they CAN vanish, due to decoherence. Nevertheless, he does not even mention decoherence - at this place at which a "Modern Introduction" to QM should.