No they don't

Of course they do. Apoptosis is fundamental to all multicellular creatures, even plants. It is essential from zygote to ultimate organismal demise for every creature on this planet.

Population stagnation and genetic stagnation have been observed in nature countless times.

I assert that this happens when there is no need for change. Most of our adaptations are epigenetic which happens as a response to our environment on the individual scale. Some of these epigenetics are then passed on to our offspring.

Which could lead to population stagnation and limited cultural and biological growth/evolution

I assert it's not a fear, that it's a natural reality.

I should have said the implication that "population stagnation", and "limited biological evolution" are bad, along with your statement of "limited cultural evolution" (whatever that means) are also bad is a fear without evidential support.

On a related note, I'd like to point out a few of the species that don't naturally get cancer tend to avoid UV exposure.

The lack of cancer is not necessarily a resistance to cellular damage, but rather the ability for cells to take care of the damage. For example, one of the most highly cancer resistant species is elephants. Elephants are in the sun all the time. However, they have an extra copy of p53, which means their cells are more likely to enter apoptosis upon damage. This ease of apoptosis is what is thought to be the cancer resistant mechanism of elephants. This also suggests apoptosis inducing agents (for those cells on the cusp) would be an excellent path towards cancer prevention and therapy.

Naked mole rats, while they are not exposed to UV are thought to be protected from cancer because of a molecule called hyaluronan which adds additional structure to the extracellular matrix, preventing cells from unrestricted growth. In both of these cases the anti-cancer mechanism is something that takes care of the cancer. It doesn't prevent it.

When I say that cancer is dietary I mean something similar. It isn't that our diet is intended to prevent cellular damage, but to allow the cells (and ECM) to function at optimal capacity when cellular damage happens.