Back in the early 20th century, before the establishment of the State of Israel, my late grandfather used to grow peaches and apricots from seed. When they grow large enough, he cuts the top and grafts his favorite cultivars. He may have grafted a branch from a mature tree, or an eye. I think that back then, there were some other cultivars (Worth exploring this cultivar history!), some of which were robust enough to be propagated from seed. I guess wild varieties are the best grafting stock, which may compensate for their inferior fruit quality. The stock compensates for the weaker and less robust roots of the scion. Plus, the graft from a mature tree will give a tree that bears fruit more quickly - See this paragraph on advantages of grafting.
Most peach trees sold by nurseries are cultivars budded or grafted onto a suitable rootstock. Common rootstocks are 'Lovell Peach', 'Nemaguard Peach', Prunus besseyi, and 'Citation'. The rootstock provides hardiness and budding is done to improve predictability of the fruit quality.
Dwarfing rootstocks will yield smaller trees but they will fruit earlier. Source . Some rootstock cultivars are given.
Apple trees on dwarfing rootstocks can be planted very close together, making management faster and easier.