My inclination is that it's both: people who are gritty are more likely to be interested in (and seek out) training with hard contact, and training with any significant contact (including falls) does build toughness.
I'd argue that it's much more the first. People are likely to want to train in a style that's compatible with them, and I've learned most of the population doesn't actually want to do anything that might result in pain. So most people that look for, and more importantly continue past the first few lessons, with hard contact are those that are "gritty" enough to be okay with the level of pain. Their toughness might increase, but they've got to have that base level of toughness to begin with.
The exception is those that have some sort of motivation that is more powerful than their desire to avoid pain (whether that's honor, revenge, avoiding other pain, fear, etc.). But IME most people, even those that think they've got that motivation, don't actually when it comes to training a hard contact style.
Iāll offer my own experience here as it seems relevant.
When I started martial arts training, I was uncoordinated, unathletic, and very timid about physical contact and getting hit. I literally could not hold my own in a pillow fight.
The first martial art that I practiced seriously for a significant length of time was Bujinkan Budo Taijutsu (or āninjutsuā as it was marketed in those days). We didnāt do real sparring, but even the choreographed, compliant technique drills involved physical contact and the experience of seeing someone punching or kicking at me (even if I generally wasnāt actually getting hit hard). Every so often weād get hit or thrown a little harder than expected and I got used to that. The process gradually desensitized me to contact to the point where I could do some light sparring with friends who did other arts and not mind if I occasionally got accidentally bopped a little harder than I was used to.
Next I spent some time in the SCA and participated in their version of heavy weapons fighting. This was a significant step up in contact intensity, but there were some mitigating factors. We wore armor and heavy helmets, which meant that I experienced head hits mostly as a loud noise and hits elsewhere generally didnāt produce anything worse than a bruise. In addition, the rules meant that we didnāt have to stand up to continuous damage. One clean hit to an arm or leg meant that limb was out of commission. One clean hit to the head or body meant that you were ādeadā and and lost the match. Still, I got used to getting hit harder than I had been in the Bujinkan, and I got more experience with the mental pressure of someone trying to hit me hard in non-compliant sparring.
After a while I moved on to Muay Thai and jiu-jitsu (first a Danzan Ryu spinoff and then BJJ). I got a fairly easy introduction to Muay Thai, because my instructor was focused on building up non-fighters rather than running a serious fight gym. That meant he eased me in to hard sparring gradually rather than throwing me in the deep end. Even so, it took me a while to adapt. My first time holding the Thai pads, I had an old pair of worn out pads and an experienced training partner who could kick hard. My forearms hurt so much from holding the pads for him that I had to grit my teeth to keep from crying.
Gradually I adjusted to that and started training with higher levels of contact and tougher opponents. Iāve competed in kickboxing, SCA, Judo, BJJ, Sumo, and HEMA. Iāve sparred professional fighters (at moderate contact levels for the most part). Iāve been knocked down, Iāve been knocked out, Iāve had broken bones and dislocations and stitches, Iāve been twisted into knots, and Iāve had countless bruises, contusions, and strains.
These days I have a strong preference for training in arts which involve sparring with a significant degree of contact. I do make allowances for the fact that Iām 57 years old, donāt heal as fast as I used to, and I donāt want to risk excessive head trauma. (So if I spar professional fighters these days, I make sure itās guys that I can trust to maintain control and not give me a concussion or put me in the ER.)
The me of 40 years ago would be terrified to do the training I do now. If you tossed that 17 year old version of myself into my current sparring sessions, he would panic and likely quit.
So I think that Gerry is right when he says the process works both ways. Tougher people are attracted to tougher training but tougher training does build tougher people. Sometimes the ramp up needs to be more gradual. Not everyone is ready to jump into full contact from the start, but I think that most people can get there eventually, given encouragement, an appropriate path for progression, and a passion for learning and improving in the martial arts.