Would not you say that one purpose of training is to give you heart. For me that is all the point of aliveness and resistance training.
If I ask you to jump a 2.00 m wall on a horse and you never ridden before, you will be scared and rightly so, after a few years of training it will be just a jump. as you said the more you train the simpler it gets.
There is no denying that 1v1 professional fighter are great athlete, just as it is obvious that whatever the competitive combat sport you are doing, it is the sum of the best particle for the activity and the rule set.
And that athleticism and the competition put you in good stead in SD.
However it is important to realise and understand the implication of that there is a different set of rule between SD and competitive fighting.
To be honest in Europe and in northern America, you average "civilian" SD customer is not, in all likeliness, going to be very skilled. So it is quite easy to build a an advantage in the skill set and competitive fighting definitely puts you there.
That being said training in jousting is not going to make me good in cattle roping, like 1v1 and SD there is a vast amount of transferable skills however what makes you a good jouster or cattle handler is not so much the skill set, it is your ability to recognise a situation and rea-act to it and you can only get that from training specifically for jousting and training specifically in cattle roping.
As kev mentioned, body conditioning should really be an important part of SD.
I understand that you can get a little tougher from training. My judo/bjj training has probably doubled my pain tolerance. But I don't consider that heart. I think heart is way above that. Yes, if you want to give me some bruises, bloody up my nose, crossface my lips to a mess, break some fingers or toes I'm fine with that. It comes with the sport. I've had and pushed though the sports injuries, the injured shoulders, the twisted fingers, broken ribs, etc. I just don't think anything I've learned in any martial art be it bjj, aikido, tkd, krav maga, judo, mauy thai, etc has taught me how to deal with a real injury. Losing teeth for example, or an eye, or breaking an arm.
I've only had one life changing injury in martial arts. That was when I broke my ankle in judo. I didn't have the heart to even try to keep fighting that match. I instead hit the ground, curled up and screamed like a little girl until the pain subsided. Not really a useful talent.