**EDIT:** here is a short, but non-elementary argument to prove that the threefold in question is uniruled:


The canonical class of your complete intersection is $(-6+2+3)H=-H$ where $H$ is the hyperplane section. Therefore this is a Fano and hence uniruled by the [Miyaoka-Mori criterion][1], which is essentially an application of Bend and Break.


  [1]: http://www.jstor.org/stable/1971387