I read that as saying that binary search isn't among those "nontrivial problem"s, along with most other things with known exact deterministic sublinear time alg

I read that as saying that binary search isn't among those "nontrivial problem"s, along with most other things with known exact deterministic sublinear time algorithms. And your first quote is followed by "However, for most natural problems", which further indicates that the known exact algorithms are for trivial problems.