*This post is going to talk about sensitive subjects that should not be taken lightly in a serious discussion. If you are easily triggered by mention of traumatic experiences, then you should stop reading now.
I don't think that's possible. I'm going to work off of the most basic definition of mental health which is "a person’s condition with regard to their psychological and emotional well-being." Creating a character with no regard to mental health would result in something less than a cardboard cutout. A person's mental health is not only defined by the traumatic moments in our lives but by the joy, as well. You can't address a character's passion for a certain subject without also considering their mental health. Whether someone is a pacifist or believes that might is right is tied into their mental health.
Take PTSD, for example. Even today with all of the knowledge and understanding that is out there concerning PTSD, I've known people who seem to think that you can't suffer from it unless you've been to war. But anybody who has had a sufficiently traumatic experience such as being witness to a suicide, killing someone, surviving assault and rape (etc) could potentially suffer from PTSD.
But since PTSD is so often associated with war veterans, let's use that as an example. Could you imagine building a character, a soldier, who has survived the horrors of war without ever addressing their mental health? It's simply not possible. Sure, you could argue that maybe they escaped with a healthy mentality. But how true do you think that is of most soldiers? Are you telling me that he doesn't have any friends from the war who have problems? Are we just going to ignore them and act like their experiences wouldn't affect him in some small way? Because even the mental health of our friends and family can affect our own mental health.
Maybe your character doesn't have any friends from the war. Maybe they all died. Do you still expect me to believe that your soldier has zero carryover from that? Am I to believe that he left a hellish warzone with zero friends left and it hasn't impacted his mental health in the slightest? That's inconceivable.
So no, I don't think mental health should be ignored in role plays because I don't think it can be without creating an uneventful world full of one-dimensional characters. However, mental health should be approached with a certain amount of care and reverence. If we plan on portraying a character with a mental health disorder, then we should dedicate some time to learning about that disorder. And if we suffer from that disorder ourselves, then we should still take the time to learn about how it affects other people because my PTSD is not Susan's PTSD and her PTSD is not yours. This goes for many other mental health disorders such as depression and schizophrenia which can manifest differently in different people.
I'm not saying that one needs to have a degree in psychology to accurately depict these characters, just that I'd like to see that they took the care and time to understand them at least a little bit. I also believe that this is more important for authors than it is for role players because among role players, some people just want to play a character and have some fun and they shouldn't be beholden to hours of research for something that is nothing more than a fun hobby.
Walk with care and write with care. Let the people around you encourage what you write but never let them tell you what not to write.