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I don't think there are any combustion reactions here (besides the source flame), since these are already fully oxidized salts. It's only internal electronic transitions of the metal ions.

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I don't know much about chemistry, so I don't really understand the significance of that.

Is the idea that there's no highly energetic reaction, and so nothing to produce the energy levels needed for intense UV?

Not my area of expertise but I believe the explanation is: the radiation profile is blackbody radiation proportional to the heat of the flame. The metal salts color the flame by absorbing light and re emitting it according to their electron transitions. However the don't add additional energy so there will be no UV light, since the original flame doesn't produce (almost any) photons that energetic. Magnesium burns much hotter and emits uv directly from blackbody profile.

It's a small fix, but the original flame emits mostly UV light. It's just almost harmless low energy UV.

The issue with magnesium burning is that both the UV light is way more energetic, and it tends to also be very intense.

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