On A Question about Quintic Del Pezzo 3-fold of Degree 5

A question from my classmate:

Question. The Quintic Del Pezzo 3-fold V(5) of Degree 5 is the intersection of \mathrm{Gr}(2,5) \subset \mathbb{P}^9 with a codimension 3 linear subspace. Show that for any point p\in V(5), there is a line not passing through that point.

Proof. I posted my answer on MO here: https://mathoverflow.net/a/354809/144294. Sasha gave a short answer in the same post using the fact that the Hilbert scheme of lines on V(5) has dimension 2 (it’s isomorphic to \mathbb{P}^2). This fact is not within my specialisation. But related to this fact, I have added a discussion that the dimension of the subvariety of \mathbb{P}^9 characterising lines on V(5) is 2 by an incidence correspondence argument.