Or even with a computer, this should have been trivial to brute-force, no? I mean, you wouldn't know that it would work ahead of time, but I find it hard to believe that no one tried it? Or am I missing some deep complexity of this problem?
On the one hand, there are infinitely many geodesics on a dodecahedron, of unbounded length, so you can't really brute force all of them.
On the other hand, the actual solution only goes through each face at most once. There are infinitely many unfoldings to try, but for each unfolding there are only finitely many paths to try, and you can get the solution on one of the minimal unfoldings.
Edit: Even simpler - there are 43,380 nets of a dodecahedron, and each has 20 vertices. So if you just draw ~8 million straight lines and see if any are both on the net and do not intersect another vertex, you'll find (at least) one that works!