Other answers here are lackluster. It’s not a question of physics; the statement you made is true if we were talking about a white nonorganic surface vs. a black nonorganic surface.
It’s a matter of biochemistry. 1. Sunburns are not due to heat, they’re due to radiation damaging cells of the epidermis. In fact, UV-A passes through windows, which will damage your skin without sunburning it (or you feeling hot). 2. Melanin’s UV-protective properties are not because of its color, but rather its molecular properties that allow it to dissipate UV radiation as heat. Chlorophyll in plants (green, not black) also isn’t bothered by UV radiation by other molecular means.
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