Does there exist a travelling salesmen generating algorithm?
For posterity and because I never received any answers in the end: my best guess is $\forall x_0 \in [0,1]^2$ was to pick any point $x_1 \in [0,1]^2$, then generate a second point $x_2$ such that the interior angle between the line between $x_0$ and $x_1$ is greater than 90$^{\circ}$ and less than 180$^{\circ}$.  Now I can see two paths forward, one is to make a sort of quasi convex structure (obviously not every solution) by alternating whether the angles are interior or exterior and staying within the bounds. The solution is then $x_{0,...,n}$ for the minimum path.

The second possibility would be to put restrictions on the distance from the last point to the next point, (say halving each time) and again requiring the angle to be constrained, but perhaps this time we can loosen the constraint to greater than 90$^{\circ}$ and less than 270$^{\circ}$ and just require that we cannot cross a line we've already made. I can't guarantee this algorithm will always generate the correct solution, however I can guarantee that it won't generate all points in $[0,1]^2$, so the question remains unfortunately unanswered.

And for the PS in my original question: this might be a good place to get started for those looking.