UV light is ionizing radiation. It's not as penetrative as gamma radiation but it still interacts in the same manor as gamma.

The interaction of UV light with the skin causing a sunburn is the skin being damaged by ionizing radiation ultimately causing it to die off. Skin cancer happens when the DNA of a skin cell gets damaged in a way that doesn't cause the cell to die and instead reproduces uncontrolled.

That's the same thing that happens when someone is exposed to ionizing radiation from nuclear waste/fallout. The main difference being that damage isn't limited to the skin, it penetrates deeper due to the higher frequency of gamma radiation.

It's really not disingenuous. It's just something people don't think about because getting a sunburn is a common human experience.