Why does lattice enthalpy decrease with increasing ionic size?
Think about Coulomb's inverse-square law:
$$F = k\frac{q_1q_2}{r^2},$$
where $q$ is the magnitude of the charge, $r$ is the distance, and $k$ is the Coulomb’s constant.
Mathematically speaking, as $r$ increases, the magnitude of $F$ becomes smaller. As far as the charge density, as it increases, the magnitude of $F$ increases.
So both explanations are saying the same thing from different point of views. As ionic radii increases, charges are further apart so attraction decreases. As charge density increases, the attraction increases since the charges are increasing in magnitude.
Just as $\ce{MgO}$ is stronger then $\ce{NaCl}$ since $\ce{Mg}$ and $\ce{O}$ have smaller ionic radii and they both bare a $2+$ and $2-$ charge, respectively, making the lattice enthalpy greater.