I think the meaning of the term "exist" needs to be clarified. All of the examples you describe except the Ramsey-theoretic one depend on axioms independent of ZF (e.g. the ultrafilter lemma). On the other hand, the probabilistic method can prove, in ZF, that plenty of objects exist (e.g. efficient sphere packings, families of graphs realizing bounds on the Ramsey numbers) for which we do not have deterministic constructions. I assume this is what Harrison is referring to (the use of the probabilistic method in Ramsey theory).

I think the meaning of the term "exist" needs to be clarified. All of the examples you describe except the Ramsey-theoretic one depend on axioms independent of ZF (e.g. the ultrafilter lemma). On the other hand, the probabilistic method can prove, in ZF, that plenty of objects exist (e.g. efficient sphere packings, families of graphs realizing bounds on the Ramsey numbers) for which we do not have efficient deterministic constructions. I assume this is what Harrison is referring to (the use of the probabilistic method in Ramsey theory).