I don't mean to split hairs but the concept of suffering needs to be delineated a lot in order for this discussion to be effective.
Suffering and pain are not the same thing. Being alive means you experience pain, also known as negative stimulation, in order to understand some of the consequences of one's environment, and of living. Pain is unavoidable.
Suffering is optional because it relates to how mind is used to frame the pain that's happening. It's like how in one instance someone can punch you in the arm lovingly to get a rise out of you, and in another case they're doing it because they hate you and want to hurt you. In one case you might suffer less, and in the other you might suffer more, even though the strength of the stimulation is the same. The only difference here is mind. One of the reasons why torture is not an effective means of truthful extraction of information is that people become desensitized to the pain through dissociation. So really, suffering is only a product of mind. If there's no mind, then suffering does not happen, even if the organism experiences constant pain. As organisms we instinctively move away from pain out of self-preservation, but we can suffer long after the pain is gone due to mental attachment.
Suffering in the existential sense refers to the constant state of "dissatisfactoriness" that everyone experiences, that nothing outside of oneself seems to quench for very long. Things that make us happy or pleased never seem to last forever and we can easily become trapped in the cycle of desire. Everything is temporal and there's nothing we can do to prevent change. In this case, we suffer because of attachment to things, instead of recognizing their temporal nature. Again, it relates to mind. True contentment can only come from this moment, and nothing beyond that. It is a choice and a practice, not something that's 'out there' to be found.