Following the approach in the question, I give an answer here using only elementary algebraic geometry and the fact that there is a line contained in $V(5)$.
We know that the Grassmannian $\operatorname{Gr}(2,5)$ is defined by $5$ linearly independent quadrics in $\mathbb{P}^9$ and $V(5)$ can be viewed as defined by $5$ linearly independent quadrics in $\mathbb{P}^6$.
Given a point $p=[1,0,\dots,0]$, suppose $V(5)$ is defined by $Q_i, 1\leq i\leq 5$, where $Q_i$ are homogeneous of degree $2$.
Write $$Q_i= q_{i,2}(x_1,\dots, x_6)+ x_0 q_{i,1}(x_1,\dots, x_6)$$
with $q_{i,k}$ homogeneous of degree $k$.
I claim that
there there is a line contained in $V(5)$ passing through $p$ if and only if $$q_{i,k}=0, \quad 1\leq i \leq 5, k=1,2 \quad (*)$$ have a common zero in $\mathbb{P}^{5}=\mathbb{P}(k[x_1,\dots x_6])$. To see this, note that given a common zero $[a_1,a_2,\dots,a_6]\in\mathbb{P}^{5}$, the line joining $p$ to $[0, a_1,a_2,\dots,a_6]$ is contained in $V(5)$ and the line joining $[1,0,\dots,0]$ and $[1,-a_1,\dots,-a_n]$ is the same as the line joining $[1,0,\dots,0]$ and $[0, a_1,\dots,a_n]$. We are given that there is at least one line contained in $V(5)$, now let $p$ be one of the points lying on this line.
Since $Q_i$ are linearly independent, we see that the linear part $q_{i,1}, 1\leq i\leq 5$ are linearly independent. By [11.3.2, Vakil’s notes](Krull’s Principal Ideal Theorem in geometry), we have that the dimension of the subvariety in $\mathbb{P}^5$ defined by $(*)$ is $\leq 0$, thus $=0$ as it's nonempty. This means $(*)$ has $k$ common zeros for $k$ a positive integer, that is, there are $k$ lines passing through $p$. I claim that for any point $q=[1, b_1,\dots,b_6] \in V(5)$ with $[b_1,\dots,b_6]$ not a common zero of $(*)$, there are also $k$ lines passing through $q$ and none of them passes through $p$. To see this, we perform a coordinate change that fixes $x_0$ and takes $q$ to $[1,0,\dots,0]$. From the above argument, we see that the lines passing through $q$ correspond to the common zeros of the $q_{i,k}$ in $(*)$ after the coordinate change. And the number of common zeros is invariant under a linear coordinate change.
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**Remark.**
Continuing the above argument, we can show that the subvariety in $\operatorname{Gr}(2,7)$ charactersing lines on $V(5)$ has dimension $\geq 2$. Consider the following incidence correspondence, from the above proof, we see that for any $x\in V(5)$, $\operatorname{dim} \operatorname{pr}_1^{-1} (x) \leq 0$ and $\operatorname{dim} \operatorname{pr}_1^{-1} (y) =0$ for any $y$ in the dense open set $\{x_0\neq
0\}$ in $V(5)$. Thus for any $p\in V(5)$ the fibre $\operatorname{pr}_1^{-1} (p)$ has dimension $0$.
[![enter image description here][1]][1]
By Upper Semicontinuity of Fibre Dimension, we have $\operatorname{dim} \Phi= \operatorname{dim} V(5)=3$. Since there exists a line $l\in V(5)$, $\operatorname{dim}pr_2^{-1}([l])=1$, by Upper Semicontinuity again, $\operatorname{dim}\Phi -\operatorname{dim} pr_2\Phi \leq 1$ thus $\operatorname{dim} pr_2\Phi \geq 2$. Now we conclude that the subvariety of $\operatorname{Grass}(2,7)$ characterising lines on $V(5)$ has dimension $\geq 2$.
A stronger result than $\operatorname{dim }pr_2\Phi\geq 2$ is that the Hilbert scheme of lines on $V(5)$ is isomorphic to $\mathbb{P}^2$ as in Sasha's answer and the reference is [Theorem 1, 2]. And the number of lines passing through a point is $3$ when counted with multiplicity [P.26, 1].
$[1]$
<cite authors="Takagi, Hiromichi; Zucconi, Francesco">_Takagi, Hiromichi; Zucconi, Francesco_, [**Geometries of lines and conics on the quintic del Pezzo 3-fold and its application to varieties of power sums**](http://dx.doi.org/10.1307/mmj/1331222846), Mich. Math. J. 61, No. 1, 19-62 (2012). [ZBL1262.14048](https://zbmath.org/?q=an:1262.14048).
$[2]$</cite><cite authors="Furushima, Mikio; Nakayama, Noboru">_Furushima, Mikio; Nakayama, Noboru_, [**The family of lines on the Fano threefold \(V_ 5\)**](http://dx.doi.org/10.1017/S0027763000001719), Nagoya Math. J. 116, 111-122 (1989). [ZBL0731.14025](https://zbmath.org/?q=an:0731.14025).</cite>
[1]: https://i.sstatic.net/M6cjS.png