I don't think it does.
Based on my analysis of how a binary search would work on this problem you would need to make a linear pass to build a sum array and then use a binary search from there. O(n log n) sort + O(n) build array + O(log n) binary search. This is more work than just
The other option is to do the binary search on the cut height. Then you would need a linear pass to calculate the sum cut. max height mh = 10^9, O(n log mh). This is the best binary search option. It's only a constant factor slower based on the constraints.
My code for both the linear search and the binary search took about the same amount of time to write and had almost exactly the same amount of lines. The only major difference was that with a little bit of care I was able to avoid the use of long long with the second binary search option which is a huge plus.