My question's about "cheap" horse keeping came from hearing about this horse that lived without ever seeing a vet (or farrier), he was born without anyone's help, and he lived the life of a wild horse, and he still lived to be 35 years old. How would things have to be done to not have vet visits and the farrier visits?
There are always those "stories" about people or things, but just because there is one exception doesn't mean yours will be to. Like the people who drink alcohol and smoke excessively all their lives and lived to over 100 - yes it might of happened, but that doesn't mean that smoking and drinking is good for you.
Most wild horses generally don't live as long as captive ones. Nor are many of them in a condition suitable for riding. They also generally have a MUCH larger area to feed from, and cover a lot more distance, wearing down their hooves. To have a captive horse in similar conditions would be near impossible - they wouldn't be close enough to catch regularly and may as well just be wild.
Vet visits are not so essential, as said earlier, you can do the shots yourself. I never get the vet out. I've had a vet out maybe twice over a period of 7+ years of owning horses. I only get them out if there is a problem I can't handle. But you still need to have the funds available should your horse be injured.
Farrier - you can learn to do it yourself, but you will still need to purchase all the tools and learn correctly, and even then I'd still get a farrier out semi-regularly to check it all out. To me, because of the strain on your body, and the cost of learning and tools, it's still better for me to pay someone else.
I do horse ownership on the cheap - but I still buy good quality food, rug them well and get all the care they need. I save money by boarding somewhere with less facilities, full time paddock rather than stabling. I won't buy Thoroughbreds because they always seem to end up costing more. I don't buy expensive gear or go to shows. I do not cheap out on feed and care though.