I'm guessing it's more of the latter. I really only know about human animals, but we probably make about 50-100 cancerous cells daily (I'm making up the number). Any one of those can become real Cancer^TM if left alone, but our immune system is so efficient at finding and killing these that cancers really only succeed once they have enough good camouflage to masquerade as normal non cancer cells and fool the immune system until it's too late. That's why transplant patients and people with autoimmune diseases are at higher risk of cancer (since they are on immunosuppressants) and boosting the immune system is an effective anti cancer therapy.

That being said honestly I doubt the immune system is hunting down tiny mutations in eggs/sperm or committing entire animal-level suicide after they are successfully fertilized but it's just that it's a harsh environment and they are perfectly evolved to survive there. Any deviation from that perfect form is just so much less fit for survival that their genes never get passed on. And other animals will eventually just become crabs. Like sharks. Every ocean will evolve crabs and sharks.