You can look at the short paper by Conte, Marchisio end Murre On the k-unirationality of the cubic complex (2007).
It contains a proof of the unirationality of $V_6$ over a field $k$ of any characteristic $\neq 2,3$, under the assumption that $V_6$ has a $k$-rational point $p$ and that one of the two planes through $p$ on the quadric is also rational over $k$.
The paper is freely available on
In the webintroduction, the authors write "we follow closely Enriques construction, our only contribution being to fully explain and justify his statements".
