I wonder if it could be solvable by an algorithm that was capable of operating at a higher level.

Mayhaps the issue is approaching the problem at the same dimensional level that the problem originated from. After all, humans can come at the problem from a higher cognitive level and, given enough time, resolve that an infinite loop is taking place, so the problem is not fundamentally unsolvable.

Einstein said, "No Problem Can Be Solved From The Same Level Of Consciousness That Created It", so it is feasible to assume that, say, a quantum computer running every possible iteration of an algorithm could easily identify which iterations of the algorithm's parameters would induce infinite loops by simple attrition.