While this is an interesting theoretical result, we need to remember that they found an algorithm that is (log n)^O(n). In other words, this is not practical to solve problems with moderate to large size n.
> In other words, this is not practical to solve problems with moderate to large size n.
This depends entirely on your definition of "moderate to large". Many real world problems can be solved easily using existing MILP solvers. We will likely never find an algorithm that can solve arbitrarily large instances of NP-complete problems in practice. Heck, it's easy to generate lists that are to large to be sorted with O(n^2) bubblesort.
I don't know much about the specific space of ILP, but speaking more generally...
It is sometimes possible to specialize algorithms and implementations to be faster for certain subdomains of the overall problem, allowing real-world-useful problems to be solved in reasonable time despite the generalized theoretical complexity bound.
If this new algorithm is a fundamentally different approach from the current ones, this may allow ILP to be used in domains where it is currently infeasible. Vice versa, this new algorithm may not be feasible in domains where current tools thrive.
