It's not the correct time to prune peaches indeed. Other than that, I don't know much about peaches and how vigorous/how much pruning they can handle, but there's a chance you might loose the tree if you cut it that hard (not all woody perennials react the same way to such harsh treatment).
On a side note: never, EVER plant a woody perennial in bought garden soil (which I think you meant is actually bark based potting compost?), only ever use your native soil and don't amend it with anything (no compost, no fertellizer, nada). These are outdated practices.
Second, just because you didn't see any root binding (bounding?
) doesn't mean there aren't any root problems further inside the root mass like circling/girdling roots which are a death sentence to woody perennials.
Below is a photo of what happens alot in pot grown shrubs and trees: they get started out as seedlings/cuttings in p9's (or whichever) and every once in a while are potted on to the next size up without any care in the world (read: no root assessment or correction nor the right depth). Result: circling and eventually girdling roots that more often than not don't outgrow their initial pot size and you'll end up with a dead shrub or tree.
The new, better way to plant woody perennials is to wash off all potting medium and to inspect their root system for defects. Cut off everything slightly wrong and plant in your native soil AT THE RIGHT DEPTH. That means the soil line should be where the trunk widens into the roots (root flare). Planting too deep suffocates the roots which again leads to death.
Anywhow, that was my rant to you. Good luck.
