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 distribution of the shortest path through $n$ points**

The following 3 diagrams plot the empirical distribution of the shortest path through $n$ points, when $n = 3, 20$ and 50 (each simulated 100,000 times). The vertical red line denotes  $0.71 \sqrt{n}$ on each plot. 

* $n = 3$:

<img src="http://www.tri.org.au/se/nequal3plot.png">

* $n = 20$:

<img src="http://www.tri.org.au/se/nequal20plot.png">

* $n = 50$:

<img src="http://www.tri.org.au/se/nequal50plot.png">


The following diagram compares:

* Marks (1948) 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: $.71 \sqrt{n}$




<img src="http://www.tri.org.au/se/mahalanobisetimateexpectedshortestpath.png">