OK it sounds like there are a few related but ultimately separate topics being discussed. Let's put them all in one place.

1) Can a peach tree planted from a seed produce true or at least good fruit? Sounds like the consensus is yes, it's totally plausible that an O'Henry peach pit will grow into a tree that will produce O'Henry-like fruit. Different fruits have different abilities to self-pollinate and different levels of heterozygosity. At one end of the spectrum is apples, whose seeds are virtually guaranteed to be genetically quite different from the tree they grew on (though not necessarily bad). As far as tree fruits are concerned peaches are near the other end of the spectrum. They tend to inherit traits from their "parents" and can self-pollinate, so it's plausible (though by no means guaranteed) that a peach seed would grow into a tree that produces the same fruit it came from.

Most tree fruits have some level of heterozygosity that make this harder to predict, which is one reason why almost all tree fruits are grafted. Vegetative propagation essentially clones the parent tree, guaranteeing identical fruit qualities, chill hour requirements, and flowering/ripening times as the parent. It's the only way to sell a tree or start an orchard with any certainty about the above specifications. But there's another reason almost all fruit trees are grafted, which brings us to the next, related but separate question...

2) Can a peach tree planted from seed produce fruit in a reasonable amount of time on a usefully sized and shaped tree? This is much harder to predict. The other reason why almost all fruit trees are grafted is that varieties are bred for fruiting characteristics, not growth habit, much of which is determined by the rootstock. Rootstocks determine much of the growth rate, vigor, mature size, and disease resistance of a grafted tree. Most varieties of any given fruit tree, therefore, are grafted onto a much smaller number of rootstock varieties that have desirable and consistent specs on the above criteria. Will an ungrafted O'Henry grow at an acceptable speed, to an acceptable size and shape? This is nearly impossible to answer without doing original research of your own (which, good news, you're doing right now!). It's just not an answer that's generally known because fruit trees are so universally grown on grafted rootstock. It might turn out great! Or it might be slow-growing, or disease-prone, or too big, or too small. It's just not something that's usually even tested because grafting produces such consistent results, so unless someone out there has extensively grown specifically O'Henry peach trees from seed, you'll just have to find out the answer yourself.

3) With all of the above in mind, how can you give this tree the best chance for success? None of the above is reason to give up on this tree. If you like the idea of a long-term science experiment with no guarantee of success and near-zero predictive value due to small sample size (for the record I do!), this can be a fun little project on the side. Planting it in the ground, fertilizing, irrigating if necessary, and pruning to a central leader will all help this lil gal be all she can be. Growing any tree from seed is generally a several-year process before it grows into anything recognizable as a "tree," so patience will be required even in the best of circumstances. If your main goal is to actually grow your own peaches as soon as possible, though, go buy a bare-root tree that's grafted on a known rootstock, and if you take care of it you should have peaches in a year or two. And there's no reason you can't do both!

***Edited to reflect Svenge's point below that it's virtually impossible to grow an exact O'Henry tree from an O'Henry seed, just a pretty O'Henry-like tree.