  Motivated by our previous work [Phys. Rev. D \textbf{104}, no.1, 016021
(2021)] on pionic leading-twist distribution amplitude (DA), we revisit
$\rho$-meson leading-twist longitudinal DA $\phi_{2;\rho}^\|(x,\mu)$ in this
paper. A model proposed by Chang based on the Dyson-Schwinger equations (DSEs)
is adopted to describe the behavior of $\phi_{2;\rho}^\|(x,\mu)$. On the other
hand, the $\xi$-moments of $\phi_{2;\rho}^\|(x,\mu)$ are calculated with the
QCD sum rules in the framework of the background field theory. The sum rule
formula for those moments are improved. More accurate values for the first five
nonzero $\xi$-moments at typical scale $\mu =1, 1.4, 2, 3~{\rm GeV}$ are given,
e.g., at $\mu = 1~{\rm GeV}$, \modi{$\langle\xi^2\rangle_{2;\rho}^\| = 0.220(6)
$, $\langle\xi^4\rangle_{2;\rho}^\| = 0.103(4)$,
$\langle\xi^6\rangle_{2;\rho}^\| = 0.066(5)$, $\langle\xi^8\rangle_{2;\rho}^\|
= 0.046(4)$ and $\langle\xi^{10}\rangle_{2;\rho}^\| = 0.035(3)$}. By fitting
those values with the least squares method, the DSE model for
$\phi_{2;\rho}^\|(x,\mu)$ is determined. By taking the left-handed current
light-cone sum rule approach, we get the transition form factor at large recoil
region, {\it i.e.} $A_1(0) = 0.498^{+0.014}_{-0.012}$,
$A_2(0)=0.460^{+0.055}_{-0.047}$, $V(0) = 0.800^{+0.015}_{-0.014}$, and the
ratio $r_2 = 0.923^{+0.133}_{-0.119}$, $r_V = 1.607^{+0.071}_{-0.071}$. After
making the extrapolation with a rapidly converging series based on
$z(t)$-expansion, we present the decay width for the semileptonic decays
$D\to\rho\ell^+\nu_\ell$. Finally, the branching fractions are
$\mathcal{B}(D^0\to \rho^- e^+ \nu_e) = 1.889^{+0.176}_{-0.170}\pm 0.005$,
$\mathcal{B}(D^+ \to \rho^0 e^+ \nu_e) = 2.380^{+0.221}_{-0.214}\pm 0.012$,
$\mathcal{B}(D^0\to \rho^- \mu^+ \nu_\mu) = 1.881^{+0.174}_{-0.168}\pm 0.005$,
$\mathcal{B}(D^+ \to \rho^0 \mu^+ \nu_\mu) =2.369^{+0.219}_{-0.211}\pm 0.011$.
