It's intuitively obvious to me that suicide served some evolutionary benefit in the past; you don't get such complex behavior in such predictable clusters without there being a reproductive benefit. As I understand, suicide has three different mechanisms each from different evolutionary pressures.
1) Negotiation suicide
2) Apoptosis suicide
3) Malfunction suicide
Negotiation suicide is attempting to get high status by threatening suicide. This is the teen suicide, jilted lover suicide, or 'cry for help' suicide. These are usually accompanied by vocal warnings and are generally done in flashy, ineffective manners. This suicide strategy is usually adopted by currently low status individuals who have the potential to be high status in the future (eg teens). The theory is that this form of suicide is a mixed strategy to achieve higher status and allotment of resources by threatening suicide. It has to be mixed strategy because if you never committed suicide you wouldn't get the increased resources (food/attention/status/etc), but if you always committed suicide you'd kill off the genes that caused it pretty quick. Unfortunately, humans are set to 'risk' suicide for status at a rate that used to be helpful 100,000 years ago, and not so much now.
Apoptosis suicide is when people sacrifice themselves so that their genes (children/family/tribe/country) can have better chances of propagation via kin selection. Note that these suicides are rarely preceded by any sort of warning and are usually done in very effective, premeditated manners. The prototypical example would be an elderly blind man with health problems choosing to die-with-dignity, and poisoning himself after having a chance to visit his grandkids one last time. Rates of suicide go up with advanced age and go up with disability. Suicide rates dramatically go up with vision impairment, which was a pretty huge hindrance in hunter-gatherer times.
The last type of suicide is just when something goes wrong in the brain. Drugs, alcohol, etc can cause this (eg Cobain). Brain damage is a cause of suicide as well, like in football players and boxers (eg Jr Seau). Bipolar, schizophrenia, and other mental illnesses also are massive risk factors in suicide, though this may arguably be apoptosis instead. Malfunction suicide could also partially account for some teen suicides, given how wild a ride hormones are.
NB: I'm working off memory of a past LW post that I can't seem to locate now on the evolutionary basis of depression and suicide, which had the actual cites.
Edit: To more directly address your point, it doesn't make sense to "say that you are too good for this game and check-out". This is almost negotiation suicide except you're using a shitty, losing strategy to do so. You only check-out if there's an advantage to be gained. It's effective to credibly threaten to check out if that can secure you more resources with the threat. The only way your brain can credibly threaten is if it's prepared to actually go all the way and some percentage of people go through with it. However, life is a game you can only win if you keep playing.
Were there really suicide methods easily available in the EEA?
I sometimes have thoughts of suicide. That does not mean I would ever come within a mile of committing the act of suicide. But my brain does simulate it; though I do try to always reduce such thoughts.
But what I have noticed is that 'suicide' is triggered in my mind whenever I think of some embarrassing event, real or imagined. Or an event in which I'm obviously a low-status actor. This leads me to think that suicide might be a high-status move, in the sense that its goal is to recover status after some event which caused a big drop in status. Consider the following instances when suicide is often considered: