I also wonder about this. Especially given how hard their lives must be. No thanks to us too- with how much we have destroyed their habitats.
I find it interesting that many suspected suicides amongst animals are those in captivity. Both from pets grieving their owners so intensely they pretty much VSED. But, also creatures like dolphins smashing their heads into their concrete enclosures.
There have been reports of tarsiers banging their heads against hard surfaces or apparently, even holding their breath until death- which I wouldn't have thought possible. Whether that is an extreme response to stress rather than a deliberate attempt to die though I suppose is harder to know.
I find it significant that animals in captivity seem more prone to possible acts of self harm and suicide. Not surprising- seeing as it's such an unnatural environment for them to be in. As it is for us. We're effectively born into captivity.
Of course, animals could be committing suicide in the wild and we simply haven't observed them. I tend to agree with you though. I'm not convinced that animals deliberately think about ending their lives the way we do.
Sometimes they do it instinctively though. A female octopus will stop eating after laying eggs and die sometimes as her babies are hatching. But, I suspect that's evolutions work- rather than a deliberate decision.
I definitely think some animals contemplate death- elephants, crows, whales, monkeys all exhibit behaviour to suggest they are curious about and even mourn death. I wonder whether that means they think about their own death though- beyond a natural instinct to avoid it.