We measure the evolution of the rest-frame UV luminosity function (LF) and the stellar mass function (SMF) of Lyman-alpha (Lya) emitters (LAEs) from z~2 to z~6 by exploring ~4000 LAEs from the SC4K sample. We find a correlation between Lya luminosity (LLya) and rest-frame UV (M_UV), with best-fit M_UV=-1.6+-0.2 log10(LLya/erg/s)+47+-12 and a shallower relation between LLya and stellar mass (Mstar), with best-fit log10( Mstar/Msun)=0.9+-0.1 log10(LLya/erg/s)-28+-4.0. An increasing LLya cut predominantly lowers the number density of faint M_UV and low Mstar LAEs. We estimate a proxy for the full UV LFs and SMFs of LAEs with simple assumptions of the faint end slope. For the UV LF, we find a brightening of the characteristic UV luminosity (M_UV*) with increasing redshift and a decrease of the characteristic number density (Phi*). For the SMF, we measure a characteristic stellar mass (Mstar*/Msun) increase with increasing redshift, and a Phi* decline. However, if we apply a uniform luminosity cut of log10 (LLya/erg/s) >= 43.0, we find much milder to no evolution in the UV and SMF of LAEs. The UV luminosity density (rho_UV) of the full sample of LAEs shows moderate evolution and the stellar mass density (rho_M) decreases, with both being always lower than the total rho_UV and rho_M of more typical galaxies but slowly approaching them with increasing redshift. Overall, our results indicate that both rho_UV and rho_M of LAEs slowly approach the measurements of continuum-selected galaxies at z>6, which suggests a key role of LAEs in the epoch of reionisation.