> There is certainly something elegant about ILP problems which gets lost when treating them with the sledgehammer that is gradient-based convex optimization.

It is funny that you call gradient-based convex optimization a "sledgehammer" since people working in combinatorial optimization (opposed to ILP) tend to denote ILP methods (e.g. cutting plane algorithms, branch & bound, branch & cut, relaxation hierarchies, ...) also as a "sledgehammer". :-D They are just jealous. :-)