Your grandmother was absolutely right, papayas have proven medicinal values (both in the flesh and the seeds), and the fruit is very good for your immune system. But that's hardly fair to expect occasional ingestion of one food to prevent all ills - cancer can come from a lot of things. If she had long-term stress in her life, was repeatedly exposed to chemicals/carcinogens, had a genetic predisposition, etc.... she could still get sick. The more things you add to your daily life that help combat that risk though, the better. Maybe without her frequent papaya eating she would have passed 10 or 20 years earlier from cancer - it is hard to know. But they absolutely do help the immune system to do what it needs to do.
I used to think like you about papayas, but now I appreciate them a lot more. I realized that a lot of the problem is eating the wrong variety, and eating them when they are overripe and stinky. Papayas are like fish - if they smell or taste really strong, they are not good anymore. I'm not terribly familiar with varieties yet, unfortunately, but I do know that here in the US, the yellow ones (often from Guatemala) are not GMO, and have a mild melon flavor that is sweet and a bit crisp, without any of the stinky feet factor when ideally ripe. I stick to yellow varieties that are not GMO.
If you are growing papaya specifically for their health benefits, it does not make sense then to add any kind of chemical fertilizer that is a carcinogen to the plant or to your land. Be careful to use only organic and safe fertilizers on your land and plants. But consider that often fertilizing a plant in the way that you have is only going to serve to feed its vegetative growth, not really help its fruit - hence why it looks so pretty but may not really produce lots of great fruit as a result.
Here in South Florida, papayas grow wild all the time. The soil is terrible with a very high PH, and especially down here in the Florida Keys we have months of drought every year, then hordes of rain. Plus, of course the storms, which make us susceptible to a lot of wind. Nonetheless, they are very happy here when left alone. They do NOT respond well to a lot of intervention - doing anything that harms their roots, or changing much of the conditions for them, easily kills them off. To me, that is not weird. Plants are all different, like animal species are all different. And many thrive much better when allowed to grow on their own without much human intervention to mess things up for them. The most I do is to intervene when starting the seeds, to give them the best chance of sprouting and surviving. After selecting the strongest ones, and giving them a little water to get through the first drought, I try to stay out of it. I'm grateful for a species that does not require so much fuss, given how many need you to be on them every minute to make sure they survive and fruit successfully!
I say once you get a few strong seedlings going in an area with good drainage where they won't get flooded or overwatered, and if anything where the PH is on the higher side rather than too low, leave them alone and see how they do. Probably you'll have better results, and you'll certainly have more time on your hands!
Incidentally, though, we have had a number of threads talking about the health benefits of various fruits and vegetables, and if you are interested in plants with strong benefits, or specifically cancer-fighting plants, there are quite a few that are sort of medicinal miracles that you would be able to grow in your climate. Papaya is great, but there are others that are even stronger, and perhaps if you sample around you'll find species that you like better taste-wise.