Not an answer, just an illustration for $6$ points, $0 \le x+y \le 2$.
[![Poly3][1]][1]
- $A,B,C$: Point $3$ cannot connect to $1$ or $6$, so it must connect to $2,5$ or $4,5$ or $2,4$.
- $B$: Point $6$ is now isolated by $34$ from $1$ and $2$.
- $C$: Point $1$ is now isolated by $34$ from $5$ and $6$.
- $D$: $1$ is trapped.
- $E$: $2$ is trapped.
**(Later.)** Here is a simple polygonal path for the lattice points $0 \le x+y \le 6$:
[![Path6][2]][2]