We investigate the $J^{PC}=(0,2)^{++}$ $\eta_c \eta_c$ and $J/\psi J/\psi$ scatterings in $N_f=2$ lattice QCD at two pion masses $m_\pi\approx 420$\,MeV and 250\,MeV. The quark field smearing scheme used in the distillation method strongly suppresses high-momentum states, thereby limiting the maximum accessible center-of-mass energy in this study to approximately 6.6\,GeV. Given the observed near-decoupling of the $\eta_c\eta_c$ and $J/\psi J/\psi$ channels, we analyze their scattering properties using the single-channel L\"{u}scher's method. In $0^{++}$ channels, $\eta_c\eta_c$ and $J/\psi J/\psi$ have repulsive and attractive interactions, respectively, which are dominated by the quark exchange effects with the Fierz rearrangement determining the interaction characteristic. A likely $0^{++}$ virtual state is observed below the threshold by 20-40\,MeV in the ${}^1S_0$ $J/\psi J/\psi$ scattering and may have connection with $X(6200)$. In $2^{++}$ channels, the near-threshold interaction for the ${}^5S_2$ $J/\psi J/\psi$ scattering is repulsive. The di-charmonium interactions are found to be insensitive to $m_\pi$ and can be consistently understood in terms of quark rearrangement effects combined with a one-gluon-exchange mechanism, both of which are supported by our lattice QCD results. Most importantly, we observe a $2^{++}$ resonance in the ${}^5S_2$ $J/\psi J/\psi$ scattering, whose properties are $(m_R,\Gamma_R)=\big(6544(10),552(34)\big)$\,MeV for $m_\pi\approx 420$\,MeV and $\big(6539(13),546(57)\big)$\,MeV for $m_\pi\approx 250$\,MeV, which are compatible with the parameters of the fully-charmed broad structure $X(6600)$ (or $X(6400)$) reported by ATLAS and CMS.