It is not an "incorrect view of the overall cost".

Binary search talks about how long the search takes with this particular algorithm assuming the data is sorted. It does not talk at all about scenarios where the data is not sorted. In particular, it does not provide any views on costs without this assumption, let alone an incorrect one.

Yes, YOU need to consider when it is appropriate to use binary search. That is the case with all algorithms you will ever apply, and goes without saying.