The beam asymmetry $\Sigma$ for linearly polarized photons and unpolarized deuterons in coherent $\pi^0$-photoproduction on the deuteron near threshold is studied. The dependence of the results for $\Sigma$-asymmetry on the deuteron wave function of various modern $NN$ potentials and on the elementary pion photoproduction amplitude is investigated. The calculations are based on the impulse approximation in which we use the realistic and high-precision CD-Bonn $NN$ potential for the deuteron wave function and the unitary isobar MAID-2007 model for the elementary $\gamma N \to \pi N$ amplitude that describes well the threshold region. A significant role of both the deuteron wave function and the elementary amplitude on the linear photon $\Sigma$-asymmetry is obtained, in particular in the vicinity of the peak. The spin asymmetry $\Sigma$ is found to be very sensitive to the choice of the elementary amplitude.