  We revisit light-cone sum rules with pion distribution amplitudes to
determine the full set of local $B \to \pi$ form factors. To this end, we
determine all duality threshold parameters from a Bayesian fit for the first
time. Our results, obtained at small momentum transfer $q^2$, are extrapolated
to large $q^2$ where they agree with precise lattice QCD results. We find that
a modification to the commonly used BCL parametrization is crucial to
interpolate the scalar form factor between the two $q^2$ regions. We provide
numerical results for the form factor parameters -- including their covariance
-- based on simultaneous fit of all three form factors to both the sum rule and
lattice QCD results. Our predictions for the form factors agree well with
measurements of the $q^2$ spectrum of the semileptonic decay $\bar{B}^0\to
\pi^+\ell^-\bar\nu$. From the world average of the latter we obtain $|V_{ub}| =
(3.77 \pm 0.15)\cdot 10^{-3}$, which is in agreement with the most recent
inclusive determination at the $1\,\sigma$ level.
