Lets analyse the physical solution in more detail.
First construction of the graph is going to be O(E+V) as you have to tie each edge to each connected node (E is number of edges, V number of vertices on a well connected graph this will be O(E)). In terms of complexity this will be the same as constructing a graph is software, but we can already imagine that the constant factors will be much larger for string tying.
Second pulling the start and end nodes does not give you the answer instantaneously, the force must propagate along the shortest path, so we can select the answer in O(L) where L is the length of the shortest path, this still looks quite good though, and that is because the force propagates in parallel. However we still have other problems here, if there are two paths that are very similar length then it will be hard to choose between them as the finite width of your strings and knots will mean they are both under tension.
Lastly lets look at scale. All your strings have a physical mass, and maximum tensile strength. Assuming we have steel wire we have a breaking length of 25.9 km. In the worst case your shortest path is 3 nodes with all other edges/nodes hanging off the middle node so this will mean (in steel) the maximum length of all the wire in your graph will be about 25.9 km. given the high constant factor in construction this may mean there are very few or even no graphs for which it is both possible and quicker to solve the shortest path problem physically.
Converting to a computer solution, if we network V hardware nodes with E message channels such that the time to propagate a message along a channel is the weight of an edge, I think we can again see that if we can send message in parallel along all edges we can find the shortest path in O(L) time. And this I think demonstrates the key issue, you are using O(E+V) hardware to do this, if we convert this to software on O(1) hardware in the straight forward way then the algorithm becomes O(L+E+V) as we now have to simulate all our hardware in series. We can now see this is worse than the usual algorithms.