I think developing toughness is a natural part of martial arts training. Whether it's the toughness of being defeated in sparring and competition in more sport-focused arts, or the toughness of perfecting your technique and improving your attitude in more traditional arts, it's always there.
However, I think a lot of people gate-keep based on toughness when you start martial arts. This is something I feel is more common in the sport arts. It's something I was guilty of when I first started teaching Taekwondo. There was a post on r/bjj from someone who works in a job where he's expected to be presentable. He had just started BJJ, and had a few bruises on his face, which was making meetings awkward. Half of the comments were something along the lines of, "If you can't handle a couple of bruises, you're too soft, go do something else you ninny."
I think this is absolutely the wrong attitude. We should be encouraging people to start martial arts to help build that toughness. For many people, it's the reason they want to learn martial arts is to build confidence. For kids, that might be why their parents put them in.
When I first started teaching, I was really hard on new white belts. This was more about their attitude and etiquette than about mental toughness, but the two were related. I would go drill-sergeant mode on 7-year-old white belts, and then they would leave class crying. My Master pulled me into his office and explained to me that white belts are not just white belts in technique, but also in attitude, and I need to treat them appropriately. In this case, "appropriate" means teaching them the etiquette and respect instead of just punishing them and being stern with them for relatively innocent breaches.
I think the same is true of mental toughness. We shouldn't try and weed out those who are not mentally tough. Instead, we should treat mental toughness as a skill to develop, and help those folks build that confidence.