The fraction of Lyman-$\alpha$ emitters among the galaxy population has been found to increase from $z\sim 0$ to $z\sim 6$ and drop dramatically at $z>6$. This drop has been interpreted as an effect of an increasingly neutral intergalactic medium with increasing redshift, while a LyC escape fraction evolving with redshift. We report the result of a large VLT/FORS2 program aiming to confirm spectroscopically a large galaxy sample at $z\ge 6$ that has been selected in several independent fields through the Lyman Break technique. Combining those data with archival data, we create a large and homogeneous sample of $z\sim 6$ galaxies ($N=127$), complete in terms of Ly$\alpha$ detection at $>95\%$ for EW(Ly$\alpha )\ge 25\backslash AA$. We use this sample to derive a new measurement of the LAE fraction at $z\sim 6$ and derive the physical properties of these galaxies through spectral energy distribution fitting. We find a median LAE fraction at $z\sim 6$ lower than in previous studies. The main difference between LAEs and non-LAEs is that the latter are significantly dustier. Using predictions of our SED fitting code accounting for nebular emission, we find an effective Ly$\alpha$ escape fraction $fesceff(Ly\alpha )=0.23-0.17+0.36$ remarkably consistent with the value derived by comparing UV luminosity function with Ly$\alpha$ luminosity function. We conclude that the drop in the LAE fraction from $z\sim 6$ to $z>6$ is less dramatic than previously found and the effect of an increasing IGM neutral fraction is possibly observed at . Based on our derived $fesceff(Ly\alpha )$, we find that the IGM has a relatively small impact on Ly$\alpha$ photon visibility at $z\sim 6$, with a lower limit for the IGM transmission to \lya\ photons, $TIGM\gtrsim 0.20$, likely due to the presence of outflows. [abdridged]