I know they answered more than just the yes/no question but looking at that 2d diagram I find myself unable to believe a computer was needed to answer just the yes/no question.
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!
Anyway the real work that made it worth publishing has to be the other paths.
By contrast, that there are other solutions to the problem, and of 30 kinds, is interesting and non-obvious.
So, it remains a mystery what the classical problem they claimed to have solved actually was.