Algorithm to find a random Hamiltonian path in a grid?

Enough randomness is very general, You should have some benchmarks, most famous algorithm for eucleadian TSP has 3/2 approximation ( Christofides algorithm ), which uses MST (like algorithm you mentioned which is 2-approximate), and as you can see in wiki best PTAS found currently has running time depending to (n log n)^f(c,2) for c > 0 (in 2 dimentional space like your sample) with approximation of (1+1/c), and best approximation for TSP with constant factor is 3/2 - 1/500 algorithm (recently found), but all of them using logical ways, there are some random usages but it doesn't cause to leave all things to random selections. If you just want using random you can use Random Walk It's more random but see Markove Chain for better performance and randomness.

Enough randomness is very general, You should have some benchmarks, most famous algorithm for eucleadian TSP has 3/2 approximation (Christofides algorithm), which uses MST (like algorithm you mentioned which is 2-approximate), and as you can see in wiki best PTAS found currently has running time depending to (n log n)^f(c,2) for c > 0 (in 2 dimentional space like your sample) with approximation of (1+1/c), and best approximation for TSP with constant factor is 3/2 - 1/500 algorithm (recently found), but all of them using logical ways, there are some random usages but it doesn't cause to leave all things to random selections. If you just want using random you can use Random Walk, It's more random but see Markove Chain for better performance and randomness.

With a ZDD that encodes all Hamiltonian paths in that graph, you could select a random Hamiltonian path where all of the possible outcomes are equally likely. I don't think it gets any better than that.

You can start with the approach you mentioned to find a Hamiltonian path. To further randomize the solution you can start rotating edges as mentioned on the wiki. Doing this more often will make the solution more random.

Rotating a random edge N*M times keeps the algorithm in the efficient realm, while making the found Hamiltonian path more random.

I cant really gove you an answer,but what I can give you is a way to a solution, that is you have to find the anglde that you relate to or peaks your interest. A good paper is one that people get drawn into because it reaches them ln some way.As for me WW11 to me, I think of the holocaust and the effect it had on the survivors, their families and those who stood by and did nothing until it was too late.