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Why does the Schottky anomaly in the heat capacity ($C_{V}(T)$ against $T$) associated with a 2-level system with energy spacing $E$ appear at $k_{B}T\approx\frac{E}{2}$ (a precise calculation gives $k_{B}T=0.4165 E$) and not at $k_{B}T\approx E$ where the thermal energy is precisely equal to the level spacing, which would allow for the most efficient thermal excitation?

I am looking for an intuitive explanation, as I understand the mathematical framework of calculating the temperature dependence of the heat capacity and no further information is provided by standard statistical mechanics/thermodynamics textbooks...

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