Hey, I get to give another "chemist's view" on this! Quick disclamer: I'm only talking about mechanisms here and trying to extrapolate to the effects on humans. Just like in my last "chemist's view" on AGEs, I'm making no representation about our evolutionary adaptations to handle things. I'm just talking about the underlying chemistry.
I'll start at the beginning: Light that we see comes in many different "colors", we call them "red", "blue", "green", etc. Ultraviolet light also comes in may different "colors", the terms physicists use are "UVA", "UVB", "XUV", "VUV" (the atmosphere absorbs all of the XUV and VUV, so we don't encounter that in the real world). Not all sunscreen absorbs all colors of UV. They generally absorb the colors of UV that cause tanning and burning (because that's the observable that people care about), but they don't generally block the higher energy UV colors that actually cause the cancer. Because the effect of that isn't marketable, no one would know it's happening.
Without sunscreen, your skin "sees" some UV colors and starts to "tan". The tanning is your body's natural protection against the higher energy UV colors that cause the cancer. So as long as you're not out long enough to get burnt (actual skin damage), the tan protects you against the bad stuff.
So strike 1 against sunscreen: it stops the good stuff that promotes tanning and lets through the bad stuff that causes cancer (that our tan would protected us against if it was allowed to form).
Strike 2 against sunscreen: The UV that causes us to tan is the same color of UV that is used in the formation of Vit D which also protects against cancer.
Now, lets say that there was a magical sunscreen that blocked 100% of all colors of UV, would I feel comfortable using it? No. Here's why:
Sunscreen works by "absorbing" UV rays. How does it work. Well it's generally a long polycyclic aromatic with lots of conjugated double bonds. The UV light is resonate with the electronic transitions of the conjugated states. When the UV light hits the sunscreen molecule, it promotes an electron into a higher state which actually breaks one of the double bonds. Most of the time the sunscreen molecule will then shed that extra energy it just absorbed as heat and reconnect that double bond. But sometimes it won't and you'll be left with a free radical (just like you're probably aware - since you're reading paleohacks - PUFAs are susceptible to oxidative damage - just image a sunscreen molecule as a more reactive PUFA). So when you started, you put a reasonably harmless chemical on your skin (or else the FDA wouldn't allow it to be sold), but it's interaction with light turns it into a potential carcinogen just like PUFAs and PAHs.
Strike 3 against sunscreen: The act of doing it's job (absorbing UV) turns it into a potential carcinogen which is now slathered over your largest organ.
Personally, ever since I worked through these mechanisms, I've never used sunscreen again.
That enough chemist rant for now. I'm happy to add to it, just don't want to bore you to death just yet.