I think the underlying issue is that the technical conditions guaranteeing for existence and uniqueness for ODEs (the Picard-Lindelof theorem) are so easy to satisfy (which is what guarantees that different numerical algorithms will give the same answer) that they're something most students are unlikely to encounter in practice.
That said, I do think there is some pedagogical value in teaching existence/uniqueness even though the result may not be so interesting because it shows students that it's possible to get information about solutions directly from the equation even without explicit formulas available. It also introduces them to the sort of abstract arguments at the core of modern mathematics.
There is definitely pedagogical value. This issue is covered by the author, but when studying ODEs for the first time, at some point the student will come across the fact that exp(x) is a solution to y' = y, which is easy enough, but needs to be convinced that exp(x) is the solution up to linear combinations. To most students this is not at all obvious! Lack of explanation here is doing the student a disservice.
That said, I do think there is some pedagogical value in teaching existence/uniqueness even though the result may not be so interesting because it shows students that it's possible to get information about solutions directly from the equation even without explicit formulas available. It also introduces them to the sort of abstract arguments at the core of modern mathematics.