The question states asymptotic results, namely that as $n$ becomes large, the shortest path $L_n$ through $n$ points satisfies
$$L_n \to \sqrt{n} \beta \quad \text{ where } \quad \beta \approx 0.71~$$
Based on a quick play, what does seem clear is that the stated asymptotic result of $0.71 \sqrt{n}$ does, in fact, do a very poor job for your desired purpose, when $n$ is not large, or indeed even when $n$ is moderately large, such as $n= 50$.
The following 3 diagrams (each representing the shortest path through $n$ points, simulated 100,000 times) plot the empirical distribution of the shortest path through $n$ points, when $n = 3, 20$ and 50. The vertical red line denotes $0.71 \sqrt{n}$ on each plot.
- $n = 3$:
- $n = 20$:
- $n = 50$:
Finally, the following diagram compares:
Marks lower bound for the expected shortest path: $\sqrt{\frac{1}{2}} \left(\sqrt{n}-\frac{1}{\sqrt{n}}\right)$
Mahalanobis estimate of the expected shortest path: $\sqrt{n}-\frac{1}{\sqrt{n}}$
The actual expected shortest path [ round dots ]
The OP's stated asymptote: $.71 \sqrt{n}$
